School on Probability Measures on Groups
Recent Directions and Trends
Infinite Dimensional Harmonic Analysis
and Probability
Jacques Faraut
CIMPA, Nice, France
and
Tata Institute of Fundamental Research, Mumbai, Inde
9-22, September 2002
i
In these notes we will present some recent results about harmonic
analysis on groups of the type
G=
∞
[
G(n),
n=1
where G(n) is a sequence of classical groups, with a sugbroup K of the
same type
∞
[
K=
K(n), K(n) ⊂ G(n).
n=1
One of the main problem in this harmonic analysis is to decompose a
continuous K-biinvariant function ϕ on G which is of positive type in a
sum or an integral of indecomposable ones, i.e. to establish a Bochner
type theorem. These indecomposable functions are called spherical. This
problem has already been considered by Schoenberg [1938,1942] and Krein
[1949]. In the first chapter we consider the case of K(n) = O(n),
the orthogonal group, and G(n) = O(n) n Rn the affine motion group.
Then K = O(∞) is the infinite dimensional orthogonal group and G the
infinite dimensional motion group. We give a proof of the main result in
[Schoenberg,1948]. In later chapters we present recent results about the
space of infinite dimensional Hermitian matrices,
H(∞) =
∞
[
Herm(n, C).
n=1
In that case K(n) = U (n), the unitary group, and G(n) is the semi-direct
product, G(n) = U (n) n Herm(n, C). Then K = U (∞) is the infinite
dimensional unitary group, and G = U (∞) n H(∞). These results are
due to Pickrell, Olshanski, and Vershik. The main source is the beautiful
paper
Ergodic unitarily invariant measures on the space of infinite Hermitian
matrices,
by Olshanski and Vershik [1996].
Surprisingly these results involve the theory of totally positive functions
which has been developped by Schoenberg ([1951]) without any connection
with group theory. It is an interesting aspect of this infinite dimensional
harmonic analysis that it involves many topics of classical analysis. One
can say the same about the theory of random matrices. But in fact this
infinite dimensional harmonic analysis involves probability measures on
spaces of infinite dimensional matrices, and hence belongs to the theory
of random matrices.
ii
We will study a family of special functions in one variable depending
on infinitely many parameters. They show up naturally in the harmonic
analysis of H(∞). So far I know they don’t have any name. We will call
them Pólya functions because they appeared in several Pólya’s papers.
A similar analysis has been studied on the infinite dimensional unitary
group U (∞). In [1976], Voiculescu established an integral representation
for functions of positive type on U (∞) which are central. See also
[Olshanski,2001], [Borodin-Olshanski,2001].
These notes correspond to a series of lectures given in September 2002
at the Tata Institute of Mumbai during the School Probability Measures
on Groups, Recent Directions and Trends. This School has been jointly
organized by the T.I.F.R. (Tata Institute of Fundamental Research) and
the CIMPA (Centre International de Mathématiques Pures et Appliquées).
I wish to thank the T. I. F. R. and the CIMPA for the invitation to take
part in this School, and also particularly Professors S.G. Dani, P. Graczyk,
and Y. Guivarc’h, who organized this School.
Jacques Faraut
Institut de Mathématiques de Jussieu
Université Pierre et Marie Curie & C.N.R.S.
175 rue du Chevaleret
75 013 Paris, France
[email protected]
Mathematics Subject Classification 2000: 43A35, 43A90, 60B11.
iii
CONTENTS
Chapter I Rotation invariant probability measures on R∞
1.
2.
3.
4.
Measures on R∞
Fourier transform of measures
Asymptotics of uniform measures on spheres
Invariant probability measures
1
3
4
6
Chapter II Harmonic analysis on the space of Hermitian matrices
1.
2.
3.
4.
Weyl integration formulas
Schur functions
Schur function expansions
Fourier transform of orbital measures
12
13
15
21
Chapter III Pólya functions
1. Pólya functions, definition
2. Generalized Wishart measures and Pólya functions
3. The Pólya-Laguerre class of entire functions
24
26
29
Chapter IV Totally positive functions and the theorem of Pickrell
1.
2.
3.
4.
Totally positive kernels
Totally positive functions
Theorem of Schoenberg
Theorem of Pickrell
35
39
42
44
Chapter V The Olshanski theory of spherical pairs
1. Spherical pairs
2. The multiplicative property of the spherical functions
48
54
Chapter VI Harmonic analysis on the space of infinite dimensional
Hermitian matrices
1.
2.
3.
4.
Spherical functions and ergodic measures
Convergence of Pólya functions
Ergodic measures are limits of orbital measures
Is there a Bochner type theorem for spherical pairs ?
References
59
62
69
74
81
iv
Chapter I
ROTATION INVARIANT PROBABILITY MEASURES ON R∞
We give first some preliminaries about probability measures on the
infinite dimensional vector space R∞ , and convergence of sequences of
such measures. Then, as a prototype for the infinite dimensional harmonic
analysis, we present classical results about probability measures on R∞
which are rotation invariant. These results are due to Schoenberg [1938].
We give a proof which differs from the original one, and resembles to the
proof of the theorem of Pickrell we will present in Chapter V.
1. Measures on R∞ . — The space R∞ is the set of all sequences
x = (x1 , x2 , . . .) of real numbers. One considers on R∞ the product
topology, i.e. the topology of pointwise convergence. This topology is
metrizable: it can be defined by the following distance,
∞
X
1
d(x, y) =
inf(|xk − yk |, 1).
2k
k=1
The space R∞ is complete, and separable. In fact the set
(∞)
Q
=
∞
[
Qn
n=1
of sequences of rational numbers with finite support is dense. The Borel
σ-field will be denoted by B. For every n we consider the projection
pn : R∞ → Rn , x = (xk ) 7→ (x1 , . . . , xn ),
and, for m < n, the projection
pm,n : Rn → Rm .
Notice that pm = pm,n ◦ pn .
If µ is a positive measure on the measurable space (R∞ , B), then µn =
pn (µ) is a positive measure on Rn . For m < n, clearly µm = pm,n (µn ).
The converse is the following
1
Theorem I.1.1 (Kolmogorov consistency theorem). — Assume that {µn } is a family of positive measures, where µn is a measure
on Rn , such that
pm,n (µn ) = µm .
One says that {µn } is a projective system or a consistent family. Then
there exists a unique measure µ on R∞ such that for all n,
pn (µ) = µn .
[Parthasarathy,1967], Theorem 5.1, p.144.
One says that a sequence µ(ν) of probability measures on V = Rn
or R∞ converges weakly to a probability measure µ if, for any bounded
continuous function f on V ,
Z
lim
ν→∞
(ν)
f (x)µ
Z
(dx) =
V
f (x)µ(dx).
V
A sequence µ(ν) of probability measures on R∞ converges
weakly to a
measure µ if and only if, for any n, the sequence pn µ(ν) converges weakly
to the measure µn = pn (µ). (See [Billingsley,1968].)
The infinite dimensional orthogonal group O(∞) is defined as
O(∞) =
∞
[
O(n).
n=1
We identify an orthogonal matrix u ∈ O(n) with the infinite matrix
.
u ..


g = ...

1

0
0
..



.

.
An element g ∈ O(∞) defines a homeomorphism of R∞ . A measure µ on
R∞ is O(∞)-invariant if and only if, for any n, the measure µn = pn (µ)
is O(n)-invariant.
Example: Gaussian measure. For t > 0 fixed, the family of the measures
{γt,n } (n ∈ N∗ ),
γt,n (dx) = pt,n (x)mn (dx),
2
with
kxk2
1
pt,n (x) = √
e− 2t ,
( 2πt)n
is consistent, hence defines a measure γt on R∞ (mn is the Lebesgue
measure on Rn ). In fact
Z
pt,n (x1 , . . . , xm , xm+1 , . . . , xn )dxm+1 . . . dxn = pt,m (x1 , . . . , xm ).
Rn−m
2. Fourier transform of measures. — Let µ be a bounded positive
measure on Rn . Its Fourier transform is defined by
Z
ϕ(ξ) =
e−ihx,ξi µ(dx).
Rn
It is a bounded continuous function,
Z
|ϕ(ξ)| ≤ ϕ(0) =
µ(dx),
Rn
of positive type : for ξ 1 , . . . , ξ N ∈ Rn , c1 , . . . , cN ∈ C,
N
X
ϕ(ξ j − ξ k )cj c̄k ≥ 0.
j,k=1
In fact
N
X
j,k=1
j
k
Z
ϕ(ξ − ξ )cj c̄k =
Rn
N
X
j 2
cj e−ihx,ξ i µ(dx).
j=1
Bochner theorem, I. — Let ϕ be a continuous function on Rn .
The function ϕ is the Fourier transform of a bounded positive measure if
and only if it is of positive type.
([Bochner,1959], Theorem 23.)
The Fourier transform is a powerful tool for studying convergence of
measures because of the following theorem.
Lévy-Cramér continuity theorem, I. — Let µ(ν) be a sequence
of bounded positive measures on Rn , and let ϕ(ν) be the Fourier transform
of µ(ν) . Assume that
∀ξ ∈ Rn , lim ϕ(ν) (ξ) = ϕ(ξ),
ν→∞
3
and that ϕ is continuous at 0. Then the sequence µ(ν) converges weakly to
a measure µ having ϕ as its Fourier transform.
The dual space of the topological vector space R∞ is the space
R
(∞)
∞
[
=
Rn ,
n=1
consisting of sequences with finitely many non zero elements. We consider
on R(∞) the inductive limit topology. A function f defined on R(∞) is
continuous if and only if, for every n, its restriction to Rn is continuous.
The Fourier transform of a bounded measure µ on R∞ is the function
ϕ defined on R(∞) by
Z
ϕ(ξ) =
e−ihx,ξi µ(dx).
R∞
Notice that the restriction ϕn of ϕ to Rn is the Fourier transform of
µn = pn (µ). From the theorems of Bochner (I) and Lévy-Cramér (I) for
Rn , and the theorem of Kolmogorov it follows :
Theorem I.2.1 (Bochner Theorem, II). — Let ϕ be a continuous
function on R(∞) . The function ϕ is the Fourier transform of a bounded
positive measure µ on R∞ if and only if it is of positive type.
([Schwartz,1973], Proposition 2, p.187.)
Theorem I.2.2 (Lévy-Cramér continuity theorem, II). — Let
µ(ν) be a sequence of bounded positive measures on R∞ , and let ϕ(ν) be
the Fourier transform of µ(ν) . Assume that
∀ξ ∈ R(∞) , lim ϕ(ν) (ξ) = ϕ(ξ),
ν→∞
and that ϕ is continuous at 0. Then the sequence µ(ν) converges weakly to
a measure µ having ϕ as its Fourier transform.
3. Asymptotics of uniform measures on spheres. — As an
illustration of the theorem of Lévy-Cramér we will present a classical
example of convergence of probability measures on R∞ . Let us consider
the uniform measure σr on the sphere of radius r and centre 0 in Rn , with
total measure equal to one. The Fourier transform ϕr of σr ,
Z
ϕr (ξ) = e−ihx,ξi σr (dξ),
is a radial function. It is essentially a Bessel function:
ϕr (ξ) = Jn (rkξk),
4
with
Jn (z) = Γ
n 2 n2 −1
J n2 −1 (z),
2 z
∞
X
1 z 2 m
(−1)m
.
=1+
n(n
+
2)
.
.
.
(n
+
2m
−
2)
m!
2
m=1
Fix t > 0, and let σ (k) be the uniform measure on the sphere S (k) of radius
√
kt and centre 0 in Rk ,
S (k) = {x = (x1 , . . . , xk ) ∈ Rk | x21 + · · · + x2k = kt},
considered as a measure on R∞ .
Proposition I.3.1. — As k → ∞, the measure σ (k) converges weakly
to the Gaussian measure γt whose Fourier transform is given by
− 2t ρ2
ψt (ξ) = e
2
,
ρ =
∞
X
ξn2
(ξ ∈ R(∞) ).
n=1
Proof. Let ψ (k) be the Fourier transform of σ (k) . For ξ ∈ Rn , n < k
√
ψ (k) (ξ) = Jk (ρ kt)
∞
X
km
(−1)m ρ2 m
=1+
t
.
k(k + 2) . . . (k + 2m − 2) m!
2
m=1
From
km
= 1,
k→∞ k(k + 2) . . . (k + 2m − 2)
km
≤ 1,
0<
k(k + 2) . . . (k + 2m − 2)
lim
it follows that
lim ϕ
k→∞
(k)
∞
X
(−1)m ρ2 m
(ξ) =
t
m!
2
m=0
t
2
= e− 2 ρ .
(k)
The statement means that, for every n, the projection σn = pn σ (k)
of the measure σ (k) on Rn converges as k → ∞ to the Gaussian measure
γt,n on Rn given by
γt,n (dx) = √
2
1
1
n e− 2t kxk mn (dx),
2πt
5
where mn is the Lebesgue measure on Rn . For n < k, this projection
(k)
σn = pn σ (k) can be computed:
Z
f (x)σn(k) (dx)
n
R
Z
k−n−2
Γ k2
1− k
2
−n
2
2
2
(kt)
mn (dx).
f
(x)
kt
−
kxk
=π
√
Γ k−n
kxk≤
kt
2
4. Invariant probability measures. — We come now to the
determination of the probability measures on R∞ which are invariant by
O(∞). Let µ be an even probability measure on R, and ϕ be its Fourier
transform. Then define the function ϕn on Rn by
ϕn (ξ) = ϕ(kξk).
It is a bounded function, therefore is the Fourier transform of a tempered
distribution Tn on Rn : for every function f in the Schwartz space S(Rn ),
Z
1
fˆ(ξ)ϕ(kξk)dmn (ξ).
hTn , f i =
(2π)n Rn
The distribution Tn , being radial, can be written as
hTn , f i = hτn , Mf (r)i,
where τn is an even distribution on R, and, for r ≥ 0, Mf (r) is the mean
of f on the sphere of radius r,
Z
Mf (r) = f (u)σr (du),
where σr is the uniform measure on the sphere of radius r, the function
Mf being extended as an even function on R.
Proposition I.4.1. —
For n = 2k + 1,
k
(−1)k
2k 1 d
r
µ.
τn =
1.3 . . . (2k − 1)
r dr
Proof.
We saw that the Bessel function Jn has the following Taylor expansion:
Jn (r) = 1 +
∞
X
(−1)m
1 r 2 m
.
n(n + 2) . . . (n + 2m − 2) m! 2
m=1
6
By a simple computation it follows that, for n = 2k + 1,
1 d k
J2k+1 (r) = (−1)k 1.3 . . . (2k − 1)
cos r.
r dr
Let f be a radial function in S(Rn ): f (x) = F (kxk), where F is an even
Schwartz function on R. Its Fourier transform fˆ,
Z
ˆ
f (ξ) =
e−ihx,ξi f (x)mn (dx),
Rn
is also radial: fˆ(ξ) = F̃ (kξk), with
Z
F̃ (ρ) = cn Jn (rρ)F (r)rn−1 dr,
R
where
n
cn =
π2
.
Γ n2
For n = 2k + 1,
1 d k
cos rρ,
ρ J2k+1 (rρ) = (−1) ak
r dr
2k
k
with ak = 1.3. . . . (2k − 1). Therefore
Z 1 d k
2k
k
ρ F̃ (ρ) = (−1) ak c2k+1
cos rρF (r)r2k dr,
r
dr
R
and, by integrating by parts,
Z
d 1 k
= ak c2k+1 cos rρ
r2k F (r) dr.
dr r
R
This means that ρ2k F̃ is the one dimensional Fourier transform of
ak c2k+1
d 1 k
r2k F (r) .
dr r
Let us now come back to the distributions τn :
hTn , f i = hτn , F i
Z
1
=
ϕ(kξk)fˆ(ξ)dmn (ξ)
(2π)n Rn
Z
cn
ϕ(ρ)F̃ (ρ)ρn−1 dρ.
=
(2π)n R
7
For n = 2k + 1,
Z
c2k+1
hτ2k+1 , F i =
ϕ(ρ)ρ2k F̃ (ρ)dρ
(2π)2k+1 R
Z c2k+1
d 1 k 2k
a
c
r F (r) µ(dr)
=
k 2k+1
2k
(2π)
R dr r
k
2 Z
(c2k+1 )
k
2k 1 d
= (−1) ak
µ(dr).
F (r)r
(2π)2k R
r dr
This is the statement, since
(2π)k
.
1.3 . . . (2k − 1)
c2k+1 =
Let us recall that a function Φ defined on an open interval ]α, β[
(−∞ ≤ α < β ≤ ∞) is said to be completely monotone if f is C ∞ and, for
all n ≥ 0,
(−1)n Φ(n) (u) ≥ 0 (α < u < β).
Theorem I.4.2 (Theorem of Bernstein). — A continuous function Φ on [0, ∞[ is completely monotone on ]0, ∞[ if and only if it is the
Laplace transform of a bounded positive measure ν on [0, ∞[,
Z
e−tu ν(dt).
Φ(u) =
[0,∞[
([Widder,1946], Theorem 12a.)
We are now ready to state the main result:
Theorem I.4.3 (Schoenberg). — Let Φ be a continuous function
on [0, ∞[, with Φ(0) = 1. The following properties are equivalent:
(i) The function ϕ defined on R(∞) by
ϕ(ξ) = Φ
1
2
kξk2 ,
is of positive type.
(ii) Φ is completely monotone on ]0, ∞[,
(iii) There is a probability measure ν on [0, ∞[ such that
Z
∞
e−tu ν(dt).
Φ(u) =
0
8
[Schoenberg,1938]
Proof.
By the theorem of Bernstein (I.4.2) properties (ii) and (iii) are equivalent.
Assume (iii), then
ϕ(ξ) =
Φ( 12
Z
2
t
2
e− 2 ρ ν(dt),
ρ )=
[0,∞[
where ν is a probability measure on [0, ∞[. It follows that the restriction
ϕn of ϕ to Rn is the Fourier transform of the measure
µn (dx) = fn (kxk)mn (dx) + aδ,
with
Z
fn (r) =
]0,∞[
r2
1
√
e− 2t ν(dt),
( 2πt)n
a = ν({0}).
Therefore the function ϕ is of positive type.
Assume now (i): the function ϕ,
ϕ(ξ) = Φ
1
2
kξk2 ,
is of positive type. For every n the restriction ϕn of ϕ to Rn is the Fourier
transform of a probability measure µn on Rn . The measure µn , being
radial, can be written as
Z
Z
f (x)µn (dx) =
Mf (r)µ̃n (dr),
Rn
R
where µ̃n is an even probability measure on R. By Proposition I.4.1, for
n = 2k + 1,
1 d k
(−1)k
µ̃2k+1 =
r2k
µ1 .
1.3 · · · (2k − 1)
r dr
Therefore, for every k, the restriction to ]0, ∞[ of the distribution
1 d k
µ1
r dr
is a positive measure. It follows that the restriction to ]0, ∞[ of the measure
µ1 has a C ∞ density with respect to the Lebesgue measure. This density
9
can be written h(r2 ) where h is a completely monotone function. By the
theorem of Bernstein, h is the Laplace transform of a bounded positive
measure α on [0, ∞[,
Z
h(u) =
e−su α(du),
[0,∞[
or
µ1 (dr) = f1 (r)dr + µ1 ({0})δ,
with
Z
2
e−sr α(ds).
f1 (r) =
[0,∞[
Let α̃ be the image of α through the map s 7→
ν(dt) = √
Then
1
2t ,
and define
1
α̃(dt) + µ1 ({0})δ.
2πt
Z
t
2
e− 2 ρ ν(dt).
ϕ(ξ) =
[0,∞[
Example: The Cauchy measure
For s > 0, we consider the following probability measure on [0, ∞[,
s2
3
s
νs (dt) = √ e− 2t t− 2 dt.
2π
The measures µn have densities:
µn (dx) = qs,n (x)mn (dx),
with
Z
∞
qs,n (x) =
0
By letting t =
1
u
kxk2
s
3
1
s
√
e− 2t √ e− 2t t− 2 dt.
( 2πt)n
2π
one computes easily the integral:
qs,n (x) = π −
n+1
2
Γ
n + 1
s
n+1 .
2
2
(s + kxk2 )− 2
In particular, for n = 1,
qs,1 (x) =
1
s
.
π s2 + x2
10
The function ϕ is given by
Z
∞
t
2
e− 2 ρ νs (dt)
ϕ(ξ) =
Z0 ∞
3
s2
t 2
s
e− 2 ρ √ e− 2t t− 2 dt
2π
0
−sρ
−skξk
=e
=e
.
=
According to Proposition I.4.1,
c2k+1 qs,2k+1 (r)r2k =
1 d k
(−1)k
r2k
qs,1 (r).
1.3 · · · (2k − 1)
r dr
The function qs,n (x) is the Poisson kernel for the half-space Rn × R+ :
If f is a bounded continuous function on Rn , then
Z
u(x, s) =
qs,n (x − y)f (y)mn (dy)
Rn
is the unique bounded solution of the Dirichlet problem :
∂2u
= 0,
∂s2
u(x, s) = f (x).
∆u +
This explains why the family of the measures {µn } is consistent:
Z
qs,n (x1 , . . . , xn−1 , xn )dxn = qs,n−1 (x1 , . . . , xn−1 ).
R
In fact, if the boundary function f does not depend on xn , the Dirichlet
problem reduces to a Dirichlet problem on Rn−1 .
11
Chapter II
HARMONIC ANALYSIS ON THE SPACE
OF HERMITIAN MATRICES
Let Hn = Herm(n, C) denote the space of n × n Hermitian matrices.
We consider on Hn the Euclidean inner product given by
(x|y) = tr(xy).
The unitary group U (n) acts on Hn by the isometric transformations
Tu (x) = uxu∗ .
Every Hermitian matrix x ∈ Hn is diagonalizable in an orthonormal basis,
and its eigenvalues are real; this means that x can be written
x = uau∗ ,
where u is a unitary matrix, a = diag(a1 , . . . , an ), and a1 , . . . , an are the
eigenvalues of x.
1. Weyl integration formulas. — Let Dn denote the space of
real diagonal matrices, Dn ' Rn . For x ∈ Rn or Cn , the Vandermonde
polynomial is defined by
D(x) =
Y
(xj − xk ).
j<k
Let m be the Euclidean measure on Hn , and α the normalized Haar
measure on the unitary group U (n).
Theorem II.1.1. — There exists a constant Cn > 0 such that, if f
is an integrable function on Hn ,
Z
Z
f (x)m(dx) = Cn
Hn
Dn
Z
f (uau )α(du) D(a)2 da1 . . . dan .
∗
U (n)
In particular, if f is U (n)-invariant :
f (uxu∗ ) = f (x)
12
(u ∈ U (n)),
then
Z
Z
f (x)m(dx) = Cn
Hn
f (a)D(a)2 da1 . . . dan .
Dn
There is a similar integration formula for the group U (n). If f is an
integrable function on U (n) which is central :
f (ugu−1 ) = f (g)
(u ∈ U (n))
(note that u−1 = u∗ ), then
Z
Z
1
f (t)|D(t)|2 β(dt),
f (g)α(dg) =
n!
Tn
U (n)
where Tn is the set of unitary diagonal matrices, Tn ' Tn , and β is the
normalized Haar measure on Tn : if t = diag(t1 , . . . , tn ), with tj = eiθj ,
then
1
dθ1 . . . dθn .
β(dt) =
(2π)n
These integration formulas are special cases of general Weyl formulas
which can be found, for the first in [Helgason,1962], (Chapter X, p.381),
and for the second in [Helgason,1984] (Corollary 5.16 of Chapter I).
The first one can also be found in a less general framework in [FarautKorányi,1994] (Theorem VI.2.3).
2. Schur functions. — Let f be a function on the unitary group
which is central :
f (ugu−1 ) = f (g)
(u ∈ U (n)),
then f is determined by its restriction to the subgroup Tn of unitary
diagonal matrices. Define
F (t1 , . . . , tn ) = f diag(t1 , . . . , tn ) .
Then F is a symmetric function of t1 , . . . , tn , i.e. is a function which
is invariant under the group Sn of permutations. Furthermore the map
f 7→ F is a bijection from the space of central functions on U (n) to the
space of symmetric functions on Tn .
For m = (m1 , . . . , mn ) ∈ Zn , tm denotes the corresponding monomial
mn
1
tm = tm
1 . . . tn .
Let F(Tn ) be the space of trigonometric polynomials, i.e. functions of the
form
X
p(t) =
am tm ,
m∈Zn
13
where the coefficients am are complex numbers, only a finite number of
them being non zero. The polynomial p is said to be symmetric if, for
every permutation σ ∈ Sn ,
p(σ · t) = p(t),
where σ · t = (tσ(1) , . . . , tσ(n) ), and skewsymmetric if
p(σ · t) = ε(σ)p(t)
(ε(σ) is the signature of the permutation σ). Let F0 (Tn ) denote the
space of symmetric trigonometric polynomials, and F1 (Tn ) the space of
antisymmetric ones.
For m = (m1 , . . . , mn ), m1 > · · · > mn the polynomial
m1
n t1
. . . tm
1 .. = X ε(σ)tσ·m
Am (t) = ...
. tm1 . . . tmn σ∈Sn
n
n
is skewsymmetric. In particular, for m = δ := (n − 1, n − 2, . . . , 0),
Aδ is the Vandermonde polynomial, Aδ = D. The polynomials Am , for
m1 > · · · > mn constitute an orthogonal basis of F1 (Tn ), and
Z
|Am (t)|2 β(dt) = #(Sn ) = n!.
Tn
For m1 ≥ · · · ≥ mn , the Schur function sm is defined by
sm (t) =
Am+δ (t)
.
D(t)
It is a symmetric trigonometric polynomial, and the Schur functions
constitute a basis of F0 (Tn ).
Let χm be the central function on U (n) whose restriction to Tn ' Tn
is equal to sm . The functions χm constitute a Hilbert basis of the space
of (classes of) square integrable central functions on U (n). The function
χm extends as a holomorphic function on GL(n, C), and, if mn ≥ 0, as a
polynomial function on M (n, C).
The function χm is the character of an irreducible representation
(πm , Hm ) of U (n),
χm (g) = tr πm (g).
In particular
χm (e) = dm := dim Hm .
14
The character χm satisfies the following functional equation: for x, y ∈
GL(n, C),
Z
1
χm (x)χm (y).
χm (xuyu−1 )α(du) =
dm
U (n)
If m = (m, 0, . . . , 0) (m ≥ 0), then sm (t) = hm (t), the complete
symmetric function:
X
hm (t) =
tα .
|α|=m
The generating function of the functions hm is given by
H(t, z) =
∞
X
hm (t)z
m
=
m=0
In fact
X
(zt)α =
α∈Nn
=
n
Y
1
.
1 − ztj
j=1
∞ X
X
∞
X
tα z m =
hm (t)z m
m=0 |α|=m
m=0
n X
∞
Y
αj
(ztj )
=
j=1 αj =0
n
Y
1
.
1
−
zt
j
j=1
3. Schur functions expansions. — To a function F defined on the
circle T with an absolute convergent Fourier series:
F (t) =
∞
X
∞
X
m
cm t ,
m=−∞
|cm | < ∞,
m=−∞
we associate the function f on the unitary group U (n) by
f (g) = det F (g).
It means that f is a central function: f (ugu−1 ) = f (g), and
f diag(t1 , . . . , tn ) = F (t1 ) . . . F (tn ).
Proposition II.3.1. — The Fourier expansion of f on U (n) is given
by
f (g) =
X
m1 ≥···≥mn
15
am χm (g),
with
am = det (cmi −i+j )1≤i,j≤n .
Equivalently
n
Y
X
F (ti ) =
am sm (t1 , . . . , tn ).
m1 ≥···≥mn
i=1
[Voiculescu,1976], Lemme 2.
If the Fourier series of F extends as a Laurent series in an annulus
r1 < |t| < r2 , the Fourier expansion of f extends as a Fourier-Laurent
series in the domain in M (n, C) defined by
{g = u1 diag(t1 , . . . , tn )u2 |u1 , u2 ∈ U (n), tj ∈ C, r1 < |tj | < r2 }.
Proof.
We will give two proofs of this formula.
a) Let us expand the following product
D(t)F (t1 ) . . . F (tn )
∞
X
X
p1 +δ
pn +δ
=
ε(σ)cp1 . . . cpn t1 σ(1) . . . tn σ(n) .
σ∈Sn p1 ,...,pn =−∞
The number am is, in this sum, the coefficient of the monomial
1 +δ1
n +δn
. It comes from the terms for which pi + δσ(i) = mi + δi
tm
. . . tm
n
1
or
pi = mi + (n − i) − n − σ(i) = mi − i + σ(i).
Therefore
am =
X
σ∈Sn
n
Y
ε(σ)
cmi −i+σ(i) = det (cmi −i+j )1≤i,j≤n .
i=1
b) We can also start from the integral formula which gives the Fourier
coefficient of a Fourier expansion for a central function on U (n):
am
1
=
n!
n
Y
Z
Tn
F (ti )sm (t)|D(t)|2 β(dt).
i=1
Since
sm (t) =
Am+δ (t)
,
D(t)
16
this integral can be written
Z Y
n
1
am =
F (ti )Am+δ (t)D(t)β(dt).
n! Tn i=1
Let us compute
Am+δ (t)D(t)
n
n
X
X
Y
Y
−δj
mi +δi
0
=
ε(σ)
ε(σ )
tσ(i)
tσ0 (j) .
σ 0 ∈Sn
i=1
σ∈Sn
j=1
By putting σ 0 = σ ◦ τ , and then τ −1 (j) = i, we obtain
=
XX
σ
ε(τ )
τ
n
Y
mi +δi −δτ (i)
tσ(i)
.
i=1
By integrating this gives
Z Y
n
F (ti )Am+δ (t)D(t)β(dt) = n! det (cmi −i+j )1≤i,j≤n .
Tn i=1
For the last part of this section we follow [Hua,1963], Chapter II, 1.2.
Let us consider n Taylor series
fi (z) =
∞
X
(i) m
cm
z
(i = 1, . . . , n),
m=0
which are convergent for |z| < r. For z = (z1 , . . . , zn ), with |zj | < r,
X
(i)
det fi (zj ) 1≤i,j≤n =
det cmj 1≤i,j≤n Am (z).
m1 >···>mn ≥0
In fact
det fi (zj ) 1≤i,j≤n
X
=
ε(σ)f1 zσ(1) . . . fn zσ(n)
σ∈Sn
=
=
X
∞
∞
X
X
m1
(n) mn
ε(σ)
c(1)
z
.
.
.
c
z
m1 σ(1)
mn σ(n)
σ∈Sn
∞
X
m1 =0
(n)
c(1)
m1 . . . cmn
m1 ,...,mn =0
=
X
mn =0
X
m1
mn
ε(σ)zσ(1)
. . . zσ(n)
σ∈Sn
X
ε(τ )c(1)
mτ (1)
m1 >···>mn ≥0 τ ∈Sn
17
. . . c(n)
mτ (n)
Am (z1 , . . . , zn ).
This can be written
det fi (zj ) 1≤i,j≤n = D(z)
X
am sm (z),
m1 ≥···≥mn ≥0
where
(i)
am = det cmj +δj 1≤i,j≤n .
Looking at the value at z = 0 of the series, one obtains
lim
det fi (zj ) 1≤i,j≤n
D(z)
z→0
Since
c(i)
m =
= a0 = det
(i) cδj
1≤i,j≤n
.
1 (m)
f (0),
m! i
the coefficient a0 can be written
1
(n−j)
(0) 1≤i,j≤n .
a0 = det fi
δ!
In general, for m = (m1 , . . . , mn ) with mj ≥ 0, one defines
m! = m1 ! . . . mn !.
It follows that, for every a,
det fi (zj ) 1≤i,j≤n
lim
D(z)
z→(a,...,a)
=
1
(n−j)
det fi
(a) 1≤i,j≤n .
δ!
For instance, if
fi (z) = z mi ,
then
det fi (zj ) 1≤i,j≤n = Am (z).
From the formula
(n−j)
fi
it follows that
(1) = mi (mi − 1) . . . (mi − n + j − 1),
(n−j)
det fi
(1) 1≤i,j≤n = D(m),
18
and one obtains the Weyl formula for the dimension:
dm = sm (1, . . . , 1) =
D(m + δ)
D(δ)
(by observing that D(δ) = δ!).
Consider now one Taylor series
f (z) =
∞
X
cm z m ,
m=0
and n complex numbers x1 , . . . , xn . Form the n Taylor series
∞
X
fi (z) = f (xi z) =
m
cm xm
i z .
m=0
(i)
Then cm = cm xm
i , and
mi
(i)
det cmj 1≤i,j≤n = det cmj xi 1≤i,j≤n = cm1 . . . cmn Am (x).
Therefore
det f (xi yj ) 1≤i,j≤n =
X
cm1 . . . cmn Am (x)Am (y),
m1 >···>mn ≥0
or
det f (xi yj ) 1≤i,j≤n
D(x)D(y)
=
X
cm1 +δ1 . . . cmn +δn sm (x)sm (y).
m1 ≥...≥mn ≥0
Consider the limit of the left handside as y → (a, . . . , a). Since
(n−j)
fi
(a) = xn−j
f (n−j) (xi a),
i
we obtain
lim
y→(a,...,a)
det f (xi yj ) 1≤i,j≤n
D(x)D(y)
n−j (n−j)
(xi a) 1≤i,j≤n
1 det xi f
=
.
δ!
D(x)
By specializing now to the case
f (z) = ez ,
19
cm =
1
,
m!
we obtain
Proposition II.3.2.
det exi yj 1≤i,j≤n
D(x)D(y)
X
=
m1 ≥···≥mn ≥0
1
sm (x)sm (y).
(m + δ)!
Furthermore
(n−j)
fi
and
(1) = xn−j
exi ,
i
xi
det xn−j
e
= D(x) exp(x1 + · · · + xn ).
i
1≤i,j≤n
Hence, as y → (1, . . . , 1), we obtain
X
exp(x1 + · · · + xn ) = δ!
m1 ≥···≥mn ≥0
1
dm sm (x).
(m + δ)!
Considering the corresponding central functions we obtain, for x ∈
M (n, C), the following expansion:
Proposition II.3.3.
exp(tr x) = δ!
X
m1 ≥···≥mn ≥0
1
dm χm (x).
(m + δ)!
Consider now the case
∞
X
1
zm
f (z) =
=
1 − z m=0
(|z| < 1).
Since cm = 1 for all m, we obtained
1
1
det
=
D(x)D(y)
1 − xi yj 1≤i,j≤n
X
sm (x)sm (y).
m1 ≥···≥mn ≥0
It is possible to evaluate this determinant; this is essentially the Cauchy
determinant, and
det
n
Y
1
1
= D(x)D(y)
.
1≤i,j≤n
1 − xi yj
1
−
x
y
i
j
i,j=1
(See for instance [Pólya-Szegö,1976] II, Part VII, No 3.)
20
Therefore:
Proposition II.3.4.
n
Y
1
=
1
−
x
y
i
j
i,j=1
X
sm (x)sm (y).
m1 ≥···≥mn ≥0
If we apply Proposition II.3.1 to the function
∞
X
1
F (x) =
=
hm (y)xm ,
1 − xyj
m=0
j=1
n
Y
we get
n
Y
1
=
1
−
x
y
i
j
i,j=1
det hmi −i+j (y) 1≤i,j≤n sm (x).
X
m1 ≥···≥mn ≥0
Comparing with the previous equality one obtains the so called JacobiTrudi identity:
sm (y) = det
hmi −i+j (y) 1≤i,j≤n .
4. Fourier transform of orbital measures. — For x ∈ Hn , the
orbital measure µx is defined by
Z
Z
f (uxu∗ )α(du),
f (y)µx (dy) =
U (n)
where f is a continuous function on Hn . We will determine the Fourier
transform of µx :
Z
e−i tr(yξ) µx (dy)
Z
e−i tr(uxu
µ
cx (ξ) =
=
∗
ξ)
α(du).
U (n)
Note that µ
cx (ξ) only depends on the eigenvalues of x and ξ. We will use
the following notation. For a kernel K(x, y) one defines
K
x1
y1
...
...
xn
yn
= det
21
K(xi , yj ) 1≤i,j≤n .
Let E be the kernel defined on R2 by
E(x, y) = exy .
For x, ξ ∈ Hn ,
Theorem II.4.1. —
X
µ
cx (ξ) = δ!
m1 ≥···≥mn ≥0
1
χm (x)χm (−iξ).
(m + δ)!
And, for x = diag(x1 , . . . , xn ), ξ = (ξ1 , . . . , ξn ),
1
E
µ
cx (ξ) = δ!
D(x)D(−iξ)
x1
−iξ1
...
...
xn
−iξn
.
This last formula is a special case of a formula proved by HarishChandra which gives the Fourier transform of an orbital measure for a compact semi-simple Lie group acting on its Lie algebra (see [Helgason,1984],
Theorem 5.35).
Proof.
By Proposition II.3.3, for z ∈ M (n, C),
X
etr z =
dm
m1 ≥...≥mn ≥0
δ!
χm (z),
(m + δ)!
and this series converges uniformly on compact sets.
x, y ∈ M (n, C),
Z
etr(uxu
f (x, y) : =
∗
y)
Therefore, for
α(du)
U (n)
X
=
m1 ≥...≥mn ≥0
Since
Z
δ!
dm
(m + δ)!
χm (uxu∗ y)α(du) =
U (n)
Z
χm (uxu∗ y)α(du).
U (n)
1
χm (x)χm (y),
dm
it follows that
f (x, y) =
X
m1 ≥...≥mn ≥0
δ!
χm (x)χm (y).
(m + δ)!
22
By Proposition II.3.2, if x = diag(x1 , . . . , xn ), y = diag(y1 , . . . , yn ),
1
x1 . . . xn
f (x, y) = δ!
E
.
y1 . . . y n
D(x)D(y)
As an application we will establish a formula for the Fourier transform
of a U (n)-invariant function on Hn . Let f ∈ S(Hn ) be U (n)-invariant,
and define
F (a1 , . . . , an ) = f diag(a1 , . . . , an ) .
We denote by fˆ the Fourier transform of f on Hn :
Z
ˆ
f (ξ) =
e−i tr(xξ) f (x)m(dx),
Hn
and F̂ the Fourier transform of F on Rn ,
Z
F̂ (b) =
e−i(a|b) F (a)da1 . . . dan .
Rn
Furthermore we write
F̃ (b1 , . . . , bn ) = fˆ diag(b1 , . . . , bn ) .
Proposition II.4.2.
F̃ (b) = Cn 1!2! . . . n!
∂
1
D
F̂ (b).
D(b)
∂b
Proof.
By using the Weyl integration formula (Theorem II.1.1)
Z Z
−i tr(uau∗ ξ)
ˆ
e
α(du) F (a)D(a)2 da1 . . . dan .
f (ξ) = Cn
Dn
U (n)
By Theorem II.4.1, for ξ = diag(b1 , . . . , bn ),
Z
1
a1
...
an
F̃ (b) = Cn δ!
E
D(a)F (a)da1 . . . dan
−ib1 . . . −ibn
D(−ib) Dn
Z
X
1
ε(σ)
e−i(a1 bσ(1) +···+an bσ(n) ) D(a)F (a)da1 . . . dan .
= Cn δ!
D(−ib)
Dn
σ∈Sn
By classical properties of the Fourier transform,
Z
G(b) : =
e−i(a1 b1 +···+an bn ) D(a)F (a)da1 . . . dan
Dn
1 ∂
F̂ (b).
=D −
i ∂b
Observe further that G is skewsymmetric. Finally
∂
1
D
F̂ (b).
F̃ (b) = Cn δ!n!
D(b)
∂b
23
Chapter III
PÓLYA FUNCTIONS
1. Pólya functions, definition. — Let Φ be a continuous function
on R, with Φ(0) = 1. For every n we associate to Φ a function ϕn on
Hn = Herm(n, C):
(x ∈ Hn ).
ϕn (x) = det Φ(x)
Note that H1 = R, ϕ1 = Φ, and that ϕn is the restriction of ϕn+1 to
Hn (with the natural embedding Hn ⊂ Hn+1 ). The function ϕn is U (n)invariant, and, if λ1 , . . . , λn are the eigenvalues of x,
ϕn (x) = Φ(λ1 ) . . . Φ(λn ).
We say that Φ is a Pólya function if, for all n, ϕn is of positive type. The
set of Pólya functions is stable under multiplication and closed for the
topology of uniform convergence on compact sets. Even more: if Φk is a
sequence of Pólya functions such that, for all λ ∈ R,
lim Φk (λ) = Φ(λ),
k→∞
and if Φ is continuous at 0, then Φ is a Pólya function.
For β ∈ R, the exponential function
Φ(λ) = eiβλ
is a Pólya function. In fact
ϕn (x) = eiβ tr(x)
is of positive type on Hn .
For γ > 0, the Gauss function
1
2
Φ(λ) = e− 2 γλ
is a Pólya function. In fact
1
2
ϕn (x) = e− 2 γ tr(x
24
)
1
= e− 2 γkxk
2
is of positive type on Hn .
Let Φ be a continuous function of positive type on R, Fourier transform
of a probability measure µ,
Z
Φ(λ) =
e−iλt µ(dt).
R
We associate to Φ the function ϕn on Hn by
ϕn (x) = det Φ(x).
Recall that Φ is said to be a Pólya function if, for all n, the function ϕn
is of positive type.
Proposition III.1.1. — The function Φ is a Pólya function if and
only if, for all n, the distribution on Rn
∂
D(t)D
µ ⊗ ··· ⊗ µ
∂t
is a positive measure.
Recall that D denotes the Vandermonde polynomial.
Proof.
The function ϕn is bounded, hence defines a tempered distribution. Let
Tn ∈ S 0 (Hn ) be its Fourier transform: for f ∈ S(Hn ),
Z
hTn , f i =
ϕn (x)fˆ(x)m(dx).
Hn
By the theorem of Bochner, the function Φ is a Pólya function if and only
if, for all n, the distribution Tn is a positive measure. Assume that f is
U (n)-invariant. Then fˆ is U (n)-invariant too. By using Proposition II.4.2,
with the same notation,
hTn , f i
Z
= Cn
Φ(b1 ) . . . Φ(bn )F̃ (b1 , . . . , bn )D(b)2 db1 . . . dbn
Dn
Z
∂
1
0
D
F̂ (b1 , . . . , bn )D(b)2 db1 . . . dbn
= Cn
Φ(b1 ) . . . Φ(bn )
D(b)
∂b
D
Z n
∂
= Cn0
Φ(b1 ) . . . Φ(bn )D(b)D
F̂ (b1 , . . . , bn )db1 . . . dbn
∂b
Dn
∂
0
µ ⊗ · · · ⊗ µ, F i.
= Cn hD(b)D
∂b
25
Therefore the U (n)-invariant distribution Tn is a positive measure if and
only if
∂
µ ⊗ ··· ⊗ µ
D(b)D
∂b
is a positive measure.
Corollary III.1.2. —
If the measure µ has a C ∞ density f :
µ(dt) = f (t)dt,
then the function Φ is a Pólya function if and only if, for t1 < · · · < tn ,
det f (n−j) (ti ) 1≤i,j≤n ≥ 0.
Proof. For a C n−1 -function f ,
∂
(n−j)
D
f (t1 ) . . . f (tn ) = det f
(ti ) 1≤i,j≤n .
∂t
Example
The function
1
2
Φ(λ) = e− 2 γλ
(γ > 0)
is the Fourier transform of the function
f (t) = √
1 2
1
e− 2γ t .
2πγ
We saw that it is a Pólya function. This can be checked by using Corollary
III.1.2. In fact, if
1
2
F (t) = e− 2 t ,
then
1
2
F (k) (t) = (−1)k e− 2 t (tk + · · ·),
and
1 2
2 Y
det F (n−j) (ti ) = e− 2 (t1 +···+tn ) (tj − ti ).
i<j
2. Generalized Wishart measures and Pólya functions. — We
consider the quadratic map
Q : M (n, k; C) → Hn
26
given by
Q(ξ) = ξξ ∗ .
The Wishart measure Wn,k is the image by Q of the Gauss measure on
the space E = M (n, k; C),
2
π −nk e−kξk m(dξ).
The space E is equipped with the Euclidean inner product
(ξ|η) = < tr(ξη ∗ ),
and m is the corresponding Euclidean measure. The Fourier transform of
the Wishart measure Wn,k is
Z
e−i(x|y) Wn,k (dy)
Hn
Z
−i x|Q(ξ) −kξk2
−nk
=π
e
e
m(dξ)
E
Z
− (I+ix)ξ|ξ
−nk
=π
e
m(dξ)
d
W
n,k (x) =
E
= det(I + ix)−k .
Therefore
d
W
n,k (x) = det Φ(x),
with
Φ(λ) = (1 + iλ)−k .
This shows that Φ is a Pólya function, for which
d
ϕn (x) = W
n,k (x),
µn = Wn,k .
With k = 1, and replacing Q by αQ, α ∈ R, we obtain the Pólya function
Φ(λ) =
1
.
1 + iαλ
Consider now a selfadjoint operator A on Ck , and let WA be the image
of the Gauss measure on E by the quadradic map
QA (ξ) = ξAξ ∗ .
27
Its Fourier transform is given by
Z
∗
2
nk
d
W
e−i(x|ξAξ ) e−kξk m(dξ).
A (x) = π
E
Proposition III.2.1. —
Let α1 , . . . , αk be the eigenvalues of A.
d
W
A (x) =
n Y
k
Y
j=1 `=1
1
,
1 + iα` λj
where λ1 , . . . , λn are the eigenvalues of x.
Proof.
We consider an orthonormal basis {f` } consisting of eigenvectors of A,
Af` = α` f` ,
and the canonical basis {ei } of Cn . If ξ ∈ M (n, k; C), v ∈ Ck ,
ξv =
k
n X
X
ξi` (v|f` )ei ,
i=1 `=1
and, if u ∈ Cn ,
ξ∗u =
k
n X
X
ξi` (u|ei )f` .
i=1 `=1
It follows that
(ξAξ ∗ )ij =
k
X
α` ξi` ξj` .
`=1
After integrating, one obtains
d
W
A (x) =
n Y
k
Y
j=1 `=1
1
.
1 + iλj α`
It follows that
−1
Φ(λ) = det(I + iλA)
=
k
Y
`=1
1
1 + iα` λ
is a Pólya function for which
d
ϕn (x) = W
A (x),
28
µn = WA .
The mean MA of WA is given by
Z
MA =
yWA (dy)
Hn
Z
2
=
ξAξ ∗ e−kξk m(dξ) = tr(A)I.
E
In fact
(MA )ij = π
−nk
Z X
k
2
α` ξi` ξj` e−kξk m(dξ).
E `=1
Therefore (MA )ij = 0 if i 6= j, and (MA )ii = tr(A). By shifting the
Wishart measure by − tr(A)I one obtains a probability measure WA0 with
mean 0. Its Fourier transform is given by
d0 (x) = det Φ0 (x),
W
A
where
Φ0 (λ) = eiλ tr(A) det(I + iλA)−1 .
It can be written
Φ0 (λ) = det2 (I + iλA)−1 ,
where det2 denotes the regularized determinant. This formula still makes
sense when A is a Hilbert-Schmidt selfadjoint operator, and one checks
that for every Hilbert-Schmidt selfadjoint operator the function
Φ(λ) = det2 (I + iλA)−1
is a Pólya function. Recall that, if A is a Hilbert-Schmidt selfadjoint
operator with eigenvalues αk , then
Y
det2 (I + A) =
e−αk (1 + αk ).
k
3. The Pólya-Laguerre class of entire functions. —
consider first the infinite product
F (s) =
∞
Y
eαk s (1 − αk s),
k=1
where αk is a sequence of complex numbers such that
∞
X
|αk |2 < ∞.
k=1
29
Let us
Proposition III.3.1. — The infinite product is uniformly convergent
on compact sets, and F is an entire function. Its zeros are the numbers
1
αk
For |z| ≤
Lemma III.3.2. —
(αk 6= 0).
1
2
,
|ez (1 − z) − 1| ≤ 2|z|2 .
Proof.
For |z| < 1,
∞
X
zm e (1 − z) = exp −
,
m
m=2
z
∞
∞
X
z m X |z|m
≤
≤
m
m
m=2
m=2
and, if |z| ≤
1
2
1
2
∞
X
m
|z|
≤
m=2
1
2
|z|2
,
1 − |z|
,
∞
X
z m ≤ |z|2 .
m
m=2
Furthermore, for w ∈ C,
|ew − 1| ≤ e|w| − 1 ≤ |w|e|w| .
Therefore
2
|ez (1 − z) − 1| ≤ |z|2 e|z| ,
1
and e 4 < 2.
Proof of Proposition III.3.1.
For R > 0, there exists N such that, if k ≥ N , then |αk | ≤
Therefore, if |s| ≤ R, |αk s| ≤ 12 , and by Lemma III.3.2,
∞
X
|eαk s (1 − αk s) − 1| ≤ 2R2
k=N
∞
X
1
2R .
|αk |2 .
k=N
Proposition III.3.3. — The entire function F is of order two at
most. More precisely, for every ε > 0, there exists C > 0 such that
2
|F (s)| ≤ Ceε|s| .
30
For z ∈ C,
Lemma III.3.4. —
2
|ez (1 − z)| ≤ e4|z| .
Proof.
For |z| ≤
1
2
, we saw in the proof of Lemma III.3.2 that
∞
X
z m ≤ |z|2 ,
m
m=2
2
therefore |ez (1 − z)| ≤ e|z| . For |z| ≥
1
2
,
2
|ez (1 − z)| ≤ e2|z| ≤ e4|z| .
(one used |1 − z| ≤ e|z| .)
Proof of Proposition III.3.3.
Let ε > 0. There exists ` such that
∞
X
|αk |2 ≤ ε,
k=`+1
and
|
∞
Y
eαk s (1 − αk s)| ≤ e4|s|
2
P∞
k=`
|αk |2
2
≤ e4ε|s| .
k=`+1
On the other hand, by using the inequality
|ez (1 − z)| ≤ e2|z| ,
one obtains
|
`
Y
eαk s (1 − αk s)| ≤ e2|s|
P`
k=1
|αk |
,
k=1
and there exists a constant C such that
P`
2
e2|s| k=1 |αk | ≤ Ceε|s| .
We will also need the Taylor expansion of the logarithmic derivative of
F:
∞
F 0 (s) X
αk =
αk −
F (s)
1 − αk s
k=1
∞
X
=−
=−
k=1
∞
X
αk2 s
1 − αk s
m=1
31
pm+1 (α)sm ,
where pm (α) is the m-th power sum
pm (α) =
∞
X
αkm ,
k=1
which is well defined for m ≥ 2. The Taylor series converges for |s| < a,
1
a = sup |αk |.
Around 1910, motivated by the Riemann hypothesis, there has been
an intense research activity about entire functions with only real zeros.
For instance the following striking result has been obtained by Pólya and
Schur.
Theorem III.3.5. — Let Ψ be an entire function with Ψ(0) = 1.
Then Ψ is a uniform limit on compact sets of polynomials with only real
zeros if and only if Ψ has the following form
Ψ(s) =
∞
Y
1
2
e−βs e− 2 γs
eαk s (1 − αk s),
k=1
with β ∈ R, γ ≥ 0, αk ∈ R and
∞
X
αk2 < ∞.
k=1
[Pólya,1913], [Pólya-Schur,1913], see also [Karlin,1968], Theorem 2.2,
Chapter 7, p.338.
Among other papers about the same topic:
[Jensen,1912-13], [Pólya,1915] (see also [Pólya-Szegö,1976], II, Part V, No
165 and following numbers).
One says that the entire function Ψ belongs to the Pólya-Laguerre class.
Let us consider its inverse, more precisely
Φ(λ) =
∞
1
2 Y
1
eiαk λ
= e−iβλ e− 2 γλ
.
Ψ(−iλ)
1 + iαk λ
k=1
It is a meromorphic function whose poles are the numbers
holomorphic in the strip
|=λ| < a,
1
= sup |αk |.
a
32
i
αk .
It is
Since
Φ(λ) = lim Φ` (λ),
`→∞
with
Φ` (λ) =
1
2
e−iβλ e− 2 γλ
`
Y
k=1
eiαk λ
,
1 + iαk λ
it is clear that Φ is a Pólya function by what we saw in Section III.2.
In fact one obtains in that way all Pólya functions. This is the
fundamental result by Pickrell [1991] that we will present in next chapter
(see also [Olshanski-Vershik,1996]).
33
Examples
In the following examples the Pólya function Φ is the Fourier transform
of a positive integrable function f :
Z
∞
e−iλt f (t)dt.
Φ(λ) =
−∞
1
2
Φ(λ) = e− 2 γλ
f (t) =
1 2
√ 1 e− 2γ t
2πγ
e−t
0
te−t
0
Φ(λ) =
1
1+iλ
Φ(λ) =
1
(1+iλ)2
f (t) =
Φ(λ) =
1
1+λ2
f (t) = 12 e−|t|
f (t) =
−iCλ
Φ(λ) = Γ(1 + iλ) = e
k=1
Φ(λ) =
πλ
sh πλ
=
Q∞ Φ(λ) =
1
ch πλ
=
Q∞ k=1
k=1
Q∞
1+
1+
λ2
k2
iλ
k
e
1+
−1
2λ2
(2k+1)2
−1
34
i λk
−1
if t > 0,
if t ≤ 0.
if t > 0,
if t ≤ 0.
−t
f (t) = e−e e−t
f (t) =
et
(1+et )2
f (t) =
1 1
2π ch 2t
=
4
ch2
t
2
Chapter IV
TOTALLY POSITIVE FUNCTIONS
AND THE THEOREM OF PICKRELL
There is a surprising connection between infinite dimensional harmonic analysis and the classical theory of totally positive functions.
1. Totally positive kernels. — A kernel K(s, t) defined over
an interval I ⊂ R is said to be totally positive if, for all numbers
s1 < · · · < sn , t1 < · · · < tn in I,
s1 . . . sn
K
≥ 0.
t1 . . . t n
and strictly totally positive if these inequalities are strict. Let us recall
the notation
s1 . . . sn
K
= det K(si , tj ) 1≤i,j≤n .
t1 . . . t n
Example. — The kernel defined by
n
1 if s ≥ t
K(s, t) =
0 otherwise
is totally positive.
This can be seen as follows. The entries of the matrix
K(si , tj ) , are equal to 0 or 1, increasing in each column, decreasing in
each line. Therefore its determinant is 0 unless K(si , tj ) = 1 for i ≥ j,
K(si , tj ) = 0 for i < j, which means that
t1 ≤ s1 < t2 ≤ s2 < t3 ≤ · · · ≤ sn−1 < tn ≤ sn ,
and is then equal to one.
As in Section II.3 let us consider a Taylor series
F (z) =
∞
X
cm z m ,
m=0
converging for |z| < r, and define the U (n)-invariant function fn on
Hn = Herm(n, C) by
X
fn diag(t1 , . . . , tn ) =
dm ãm sm (t),
m1 ≥···≥mn ≥0
35
where
ãm = cm1 +δ1 . . . cmn +δn ,
or
X
fn (x) =
dm ãm χm (x).
m1 ≥···≥mn ≥0
We consider the kernel K on ] − r, r[×] − r, r[ given by
K(s, t) = F (st).
Proposition IV.1.1. — If, for all n, fn ≥ 0, then the kernel K
is totally positive. And, if fn > 0, then K is strictly totally positive.
[Gross-Richards,1989]
Proof.
We saw in Section II.3 that
s1 . . . sn
K
= D(s)D(t)
t1 . . . t n
X
ãm sm (s)sm (t).
m1 ≥···≥mn ≥0
By using the functional equation
Z
1
χm (xuyu−1 )α(du) =
χm (x)χm (y),
dm
U (n)
one obtains
s1 . . . sn
K
t1 . . . t n
Z
X
= D(s)D(t)
dm ãm
U (n)
m1 ≥···≥mn ≥0
Z
= D(s)D(t)
χm (xuyu−1 )α(du)
fn (xuyu−1 )α(du),
U (n)
with x = diag(s1 , . . . , sn ), y = diag(t1 , . . . , tn ). The statement follows
from that formula.
Corollary IV.1.2. —
The kernel
K(s, t) = est
is strictly totally positive.
36
Proof.
It is a special case of Proposition IV.1.1 and II.3.3:
∞
X
zm
F (z) = e =
,
m!
m=1
z
1 tr x
e
> 0.
δ!
fn (x) =
In fact it can be shown directly (see [Polyá-Szegö,1976] II, Part V,
No 76). First show inductively on n that, for distinct real numbers
α1 , . . . , αn , and real numbers a1 , . . . , an , the function
f (t) = a1 eα1 t + · · · an eαn t ,
has at most n − 1 real zeros, if it does not vanish identically. Then show
inductively on n that, for s1 < · · · < sn , t1 < · · · < tn ,
s1 . . . sn
K
> 0.
t1 . . . t n
For that consider the function
s1 t1
e
. . . esn t1 ..
..
.
= a1 es1 t + · · · + an esn t .
f (t) = s t.
sn tn−1 1 n−1
.
.
.
e
e
esn t
...
esn t The function f does not vanish identically since
f (t) ∼ an esn t
and
an = K
(t → ∞),
s1 . . . sn−1
t1 . . . tn−1
> 0.
Furthermore
f (t1 ) = 0, . . . , f (tn−1 ) = 0.
Hence f (tn ) > 0.
Proposition IV.1.3. —
If the composition kernel
Let K and L be totally positive kernels.
Z
M (s, t) =
K(s, u)L(u, t)du
I
37
is well defined, it is totally positive.
Proof. It follows from the formula
M
s1
t1
. . . sn
. . . tn
Z
=
K
{ui ∈I|u1 <···<un }
which follows from
s1 . . . sn
M
t1 . . . t n
Z
1
s1
K
=
u
n! I×···×I
1
s1
u1
...
...
...
...
sn
un
sn
un
L
L
u1
s1
In fact
K
s1
t1
...
...
sn
tn
=
X
ε(σ)
σ∈Sn
u1
t1
...
...
...
...
n
Y
un
sn
un
tn
du1 . . . dun ,
du1 . . . dun .
K si , tσ(i) .
i=1
Therefore we can write
Z
s1 . . . sn
u 1 . . . un
K
L
du1 . . . dun
u1 . . . un
t1 . . . t n
I×···×I
Z
n
n
X
Y
Y
X
=
ε(σ)
K sσ(i) , ui
ε(τ )
L (ui , tτ (i) du1 . . . dun
I×...×I σ∈S
n
=
X
ε(στ )
σ,τ ∈Sn
= n!M
n
Y
τ ∈Sn
i=1
M sσ(i) , tτ (i)
i=1
i=1
s1
t1
. . . sn
. . . tn
.
Let K be a totally positive kernel, and f , g two positive functions.
Then the kernel L(s, t) = f (s)g(t)K(s, t) is totally positive.
Let K be a totally positive kernel, and ϕ, ψ two increasing functions,
then the kernel
L(s, t) = K ϕ(s), ψ(t)
is totally positive.
For instance, if ϕ, ψ are increasing functions, then the kernel
K(s, t) = eϕ(s)ψ(t)
38
is totally positive.
2. Totally positive functions. — A measurable function f defined
on R is said to be totally positive if the kernel
K(s, t) = f (s − t)
is totally positive.
Examples
a) Exponential function
K(s, t) = eαs e−αt .
f (t) = eαt ,
b) Gauss function
1
1
2
2
1
2
K(s, t) = e− 2 s e− 2 t est .
f (t) = e− 2 t ,
c)
f (t) =
n
1 if t ≥ 0
0 otherwise
If f is totally positive then the functions eαt f (t), f (at + b) are totally
positive too. For instance, for α ∈ R,
f (t) =
n
eαt
0
if t ≥ b
otherwise
is totally positive.
Proposition IV.2.1. —
If f and g are totally positive and
integrable, then f ∗ g is totally positive too.
Proof.
It follows from Proposition IV.1.3.
In the particular case of the convolution product, the formula which
was used in the proof of Proposition IV.1.3 specializes as
Z
1
det f (ti − uk ) · det g(uk − sj ) du1 . . . dun .
det f ∗ g(ti − sj ) =
n! Rn
Proposition IV.2.2. — If f is totally positive and of class C ∞ ,
then, for t1 < · · · < tn ,
det f (n−j) (ti ) 1≤i,j≤n ≥ 0.
39
Lemma IV.2.3 . —
If f is of class C ∞ , and if D(h) 6= 0, then
1
lim
det f (ti + εhj ) 1≤i,j≤n
ε→0 D(εh)
1
(n−j)
det f
=
(ti ) 1≤i,j≤n .
1!2! . . . (n − 1)!
[Hua,1963], Theorem I.2.4.
Proof.
Consider first the case of a polynomial f of degree ≤ n − 1:
f (t + h) = f (t) + hf 0 (t) + · · · +
hn−1 (n−1)
f
(t).
(n − 1)!
Then the matrix f (ti + hj ) can be written as a product
f (ti + hj )
 f (t ) f 0 (t )
1
1
0
f
(t
)
f
(t

2
2)
=
..
 ..
.
.
f (n−1) (t1 ) 
f (n−1) (t2 ) 

..

.
...
...
f (tn ) f 0 (tn ) . . .

1
1
...
 h1
h
...
2

..
·  ..
 n−1
.
.
h1
hn−1
2
...
(n−1)!
(n−1)!
f (n−1) (tn )

1
hn 
.. 
.
. 
hn−1
n
(n−1)!
Therefore
n(n−1)
det f (ti + hj ) = (−1) 2
1
D(h) det f (j−1) (ti ) .
1!2! . . . (n − 1)!
For a C ∞ function f , one uses a Taylor expansion:
f (t + h) = f (t) + hf 0 (t) + · · · +
hk (k)
f (t) + hk ε(h),
k!
with
lim ε(h) = 0.
h→0
By expanding the determinant
det f (ti + hj ) 1≤i,j≤n ,
40
we obtain
X
det f (ti + hj ) 1≤i,j≤n =
am (t)Am(h)+δ + R(h),
|m|≤k
with
a0 (t) =
1
det f (n−j) (ti ) 1≤i,j≤n ,
1!2! . . . (n − 1)!
and R(h) is a finite sum of the form
R(h) =
X
hm εm (h),
with
m1 + · · · + mn ≥ k,
lim εm (h) = 0.
h→0
The polynomial Am+δ (h) is divisible by D(h),
X
Am+δ (εh)
R(εh)
1
det f (ti + εhj ) =
am (t)
+
,
D(εh)
D(εh)
D(εh)
|m|≤k
and
n(n−1) R(εh)
R(εh)
= ε− 2
.
D(εh)
D(h)
By taking k >
n(n−1)
,
2
one obtains the statement.
The following converse of Proposition IV.2.4 holds:
Proposition IV.2.3. — Let f be a C ∞ function on R which is
integrable, with all its derivatives. Assume that, for all t1 < · · · < tn ,
det f (n−j) (ti ) 1≤i,j≤n ≥ 0,
then f is totally positive.
Proof.
Define
fγ = f ∗ gγ ,
where gγ is the Gauss function
gγ (t) = √
t2
1
e− 2γ
2πγ
41
(γ > 0).
Then, for t1 < · · · < tn , s1 < · · · < sn ,
det fγ (ti − sj )
Z
=
det f (ti − uk ) det gγ (uk − sj ) du1 . . . dun .
{u1 <···<un }
From Lemma IV.2.3 it follows that
det fγ(i−1) (−sj )
Z
det f (i−1) (−uk ) det gγ (uk − sj ) du1 . . . dun .
=
{u1 <···<un }
If the function
det f (i−1) (−uk ) ,
which is ≥ 0 on {u1 < · · · < un }, vanished identically, the functions
f, f 0 , . . . , f (n−1) would be linearly dependent, i.e. f would be solution
of a constant coefficient linear differential equation. But this is not
possible since f is integrable.
Since gγ is strictly totally positive, the function
det gγ (uk − sj )
is strictly positive. Therefore
det fγ(i−1) (ti )
is strictly positive too. By Theorem 2.4 of Chapter 2, p.55, in [Karlin,1968], fγ is totally positive. Since, for all t,
lim fγ (t) = f (t),
γ→0
the function f is totally positive too.
([Olshanski-Vershik], Proposition 7.8.)
3. Theorem of Schoenberg. — Recall that one says that an
entire function Ψ, with Ψ(0) = 1, belongs to the Pólya-Laguerre class if
it is a uniform limit on compact sets of polynomials with only real zeros
(see Section III.3). Such a function has a representation as an infinite
product:
∞
Y
1
βs − 2 γs2
Ψ(s) = e e
e−αk s (1 + αk s),
k=1
42
with β ∈ R γ ≥ 0, αk ∈ R satisfying
∞
X
αk2 < ∞.
k=1
The following theorem is one of the key results for the topic of these
notes.
Theorem IV.3.1 (Theorem of Schoenberg). —
totally positive function on R which is integrable, with
Z
f (t)dt = 1.
a) Let f be a
R
Then its Fourier transform Φ,
Z
e−iλt f (t)dt,
Φ(λ) =
R
is of the form
Φ(λ) =
1
,
Ψ(iλ)
where Ψ is an entire function of the Pólya-Laguerre class:
Φ(λ) =
1
2
e−iβλ e− 2 γλ
∞
Y
k=1
with
γ+
∞
X
eiαk λ
,
1 + iαk λ
αk2 > 0.
k=1
b) Conversely, if
Φ(λ) =
1
,
Ψ(iλ)
where Ψ is an entire function of the Pólya-Laguerre class, and if
γ+
∞
X
αk2 > 0,
k=1
then Φ is the Fourier transform of a totally positive function which is
integrable.
[Schoenberg,1951], see also [Karlin,1968] Chapter 7, Theorem 3.2.
43
4. Theorem of Pickrell. — Let Φ be a continuous function of
positive type on R, Fourier transform of a probability measure µ,
Z
Φ(λ) =
e−iλt µ(dt).
R
We associate to Φ the function ϕn on Hn by
ϕn (x) = det Φ(x).
Recall that Φ is said to be a Pólya function if, for all n, the function ϕn
is of positive type.
Theorem IV.4.1 (Theorem of Pickrell). —
function Φ is given as an infinite product
Φ(λ) =
1
2
e−iβλ e− 2 γλ
∞
Y
k=1
Every Pólya
eiαk λ
,
1 + iαk λ
with β ∈ R, γ ≥ 0, αk ∈ R, and
∞
X
αk2 < ∞.
k=1
[Pickrell,1991], Proposition 5.9, and also [Olshanski-Vershik,1996],
Proof.
Let us regularize the measure µ: for γ1 > 0 define
fγ1 = gγ1 ∗ µ.
The Fourier transform of fγ1 is the product
1
2
Φ0 (λ) = e− 2 γ1 λ Φ(λ).
By Corollary III.1.2, for t1 < · · · < tn ,
det fγ(n−j)
(t
)
i 1≤i,j≤n ≥ 0.
1
By Proposition IV.2.4, the function fγ1 is totally positive. By the
Theorem of Schoenberg (Theorem IV.3.1), the function Φ0 admits a
representation as an infinite product
Φ0 (λ) =
1
2
e−iβλ e− 2 γ0 λ
∞
Y
k=1
44
eiαk λ
.
1 + iαk λ
Therefore
Φ(λ) =
1
2
e−iβλ e− 2 (γ0 −γ1 )λ
∞
Y
k=1
eiαk λ
.
1 + iαk λ
It remains to check that γ0 − γ1 ≥ 0. Since
|Φ(λ)| ≤ 1
(λ ∈ R),
this follows from Proposition III.3.3: for every ε > 0, there exists C > 0
such that
∞
Y
2
−iαk λ
e
(1 + iαk λ) ≤ Ceε|λ| .
k=1
Therefore
1
2
2
e− 2 (γ0 −γ1 )λ ≤ Ceε|λ| ,
and γ0 − γ1 ≥ −2ε.
The theorem of Pickrell holds also in the case of real symmetric
matrices with a slight change. Let Φ be a continuous function on R,
with Φ(0) = 1. For every n one defines the function ϕn on Sym(n, R)
by
ϕn (x) = det Φ(x),
or equivalently, ϕn is invariant under O(n) and
ϕn diag(a1 , . . . , an ) = Φ(a1 ) . . . Φ(an )
(aj ∈ R).
Then the function ϕn is of positive type for all n if and only if it can be
written
1
∞ Y
1
eiαk λ 2
−iβλ − 2 γλ2
Φ(λ) = e
e
,
1 + iαk λ
k=1
with
β ≥ 0, γ ≥ 0, αk ∈ R,
∞
X
αk2 < ∞.
k=1
By using the inclusions
Sym(n, R) ⊂ Herm(n, C) ⊂ Sym(2n, R),
and the fact that the restriction of a function of positive type to
a subgroup is of positive type too, it follows from Theorem IV.4.1.
([Pickrell,1991], Proposition 5.12.)
45
Proposition IV.4.2. — The support of the measure µ is contained
in [0, ∞[ if and only if the Pólya function Φ can be written
Φ(λ) = e−iβ0 λ
∞
Y
k=1
with
β0 ≥ 0,
1
,
1 + iαk λ
∞
X
αk ≥ 0,
αk < ∞.
k=1
([Olshanski-Vershik,1996], Remark 2.11.)
Proof.
First, if supp(µ) ⊂ [0, ∞[, then β ≥ 0 since β is the mean of µ. The
support of µ is contained in [0, ∞[ if and only if Φ is holomorphic in the
lower halfspace =λ < 0, and
|Φ(λ)| ≤ 1
(=λ ≤ 0).
One checks easily that it holds if Φ has the given form.
Assume that Φ is holomorphic for =λ < 0, and
|Φ(λ)| ≤ 1
(=λ ≤ 0).
Then αk ≥ 0. For λ = −iν, ν ≥ 0,
1
Φ(−iν) = e−βν e 2 γν
2
∞
Y
k=1
eαk ν
≤ 1.
1 + αk ν
By Proposition III.3.3, for every ε > 0, there exists C > 0 such that
∞
Y
k=1
2
eαk ν
≥ Ce−εν .
1 + αk ν
It follows that γ = 0. For every δ, 0 < δ < 1, there exists D > 0 such
that
ex
≥ Deδx (x ≥ 0).
1+x
Therefore, for every N , and ν ≥ 0,
N
N
X
Y
−βν
D exp δ
αk − β ν ≤ e
N
k=1
k=1
46
eαk ν
≤ 1.
1 + αk ν
Hence
δ
N
X
αk ≤ β,
k=1
and, since δ < 1 and N are arbitrary,
∞
X
αk ≤ β.
k=1
By putting
β0 = β −
∞
X
αk ,
k=1
we obtain
Φ(λ) = e−iβ0 λ
∞
Y
k=1
47
1
.
1 + iαk λ
Chapter V
THE OLSHANSKI THEORY OF SPHERICAL PAIRS
This chapter follows closely [Olshanski,1990], 23.
1. Spherical pairs. — We consider a topological group G, and a
closed subgroup K. We denote by P the set of continuous functions ϕ on
G which are of positive type with ϕ(e) = 1, and which are K-biinvariant.
It is a convex set. If ϕ ∈ P, there exists a unitary representation π of
G on a Hilbert space H, and a unit vector u which is K-invariant and
cyclic such that
ϕ(g) = π(g)u|u .
Recall that a vector u ∈ H is said to be K-invariant if, for all k ∈ K,
π(k)u = u, and cyclic if the subspace generated by the vectors π(g)u
(g ∈ G) is dense in H. The following properties are equivalent
(1) ϕ is extremal in P,
(2) the representation π is irreducible.
Let HK be the space of K-invariant vectors. If dim HK = 1, then the
representation π is irreducible.
The pair (G, K) is said to be spherical if, for every irreducible unitary
representation π of G on a Hilbert space H,
dim HK ≤ 1.
If dim HK = 1, the irreducible representation π is said to be spherical.
A function ϕ on G is said to be spherical if it can be written
ϕ(g) = π(g)u|u),
where π is a spherical representation, u ∈ HK , with kuk = 1. Hence
the spherical functions are the extremal points in the convex set P.
If G is locally compact, and K compact, then the pair (G, K) is
spherical if and only if it is a Gelfand pair, i.e. if the algebra of Kbiinvariant integrable functions is commutative. In that case a function
ϕ ∈ P is spherical if and only if it satisfies the functional equation
Z
ϕ(xky)α(dk) = ϕ(x)ϕ(y),
K
48
where α is the normalized Haar measure of K. (see, for instance,
[Gangolli-Varadarajan,1988], §1.5, or [Faraut,1982], Section I.)
Let G(n), K(n) be a sequence of Gelfand pairs. One assumes that
G(n) is a closed subgroup of G(n + 1), K(n) is a closed subgroup of
K(n + 1), and K(n) = G(n) ∩ K(n + 1). Let
G=
∞
[
G(n),
K=
n=1
∞
[
K(n).
n=1
Theorem V.1.1. — (i) The pair (G, K) is spherical.
(ii) Let ϕ ∈ P. The function ϕ is spherical if and only if
Z
lim
ϕ(xky)αn (dk) = ϕ(x)ϕ(y),
n→∞
K(n)
where αn is the normalized Haar measure on K(n).
Proof.
a) Let π be an irreducible unitary representation of G on a Hilbert
space H with HK 6= {0}. We will show that dim HK = 1. The
orthogonal projection onto HK(n) can be written
Z
Pn =
π(k)αn (dk).
K(n)
Note that Pn+1 = Pn+1 Pn = Pn Pn+1 , since K(n) ⊂ K(n + 1), and
therefore HK(n+1) ⊂ HK(n) . The projections Pn strongly
converge to
K
the orthogonal projection P onto H . Since G(n), K(n) is a Gelfand
pair, for x, y ∈ G(n),
Pn π(x)Pn π(y)Pn = Pn π(y)Pn π(x)Pn .
As n → ∞ one obtains
P π(x)P π(y)P = P π(y)P π(x)P.
In fact, using Pn+m = Pn+m Pn = Pn Pn+m , one obtains
Pn+m π(x)Pn π(y)Pn+m0 = Pn+m π(y)Pn π(x)Pn+m0 .
Let m → ∞, m0 → ∞, and then n → ∞.
Let A be the algebra generated by the operators P π(x)P , for x ∈ G.
This algebra is commutative. The subspace HK is invariant under A.
49
We will show that HK is irreducible under A. Since an irreducible
representation of a commutative Banach algebra is one dimensional, it
will follow that dim HK = 1.
Assume that
H K = Y1 ⊕ Y2 ,
where Y1 and Y2 are two orthogonal subspaces of HK which are Ainvariant. Let u ∈ Y1 , u 6= 0. For v ∈ Y2 , x ∈ G,
P π(x)P u|v = 0.
This means that, for all x,
π(x)u|v = 0.
Since π is irreducible, this implies that v = 0. Therefore Y2 = {0}.
b) Let ϕ be a spherical function:
ϕ(g) = π(g)u|u),
where π is an irreducible unitary representation, u ∈ HK , kuk = 1. For
v ∈ H,
(P π(g)P u|v) = (π(g)u|P v),
and, since P v = (v|u)u,
= (u|v) π(g)u|u = ϕ(g)(u|v).
Therefore
P π(g)P u = ϕ(g)u,
P π(x)P π(y)P u = ϕ(x)ϕ(y)u,
π(x)P π(y)u|u = ϕ(x)ϕ(y).
Since Pn → P strongly,
ϕ(x)ϕ(y) = lim π(x)Pn π(y)u|u
n→∞
Z
= lim
ϕ(xky)αn (dk).
n→∞
K(n)
c) Let ϕ ∈ P, and assume that
Z
lim
ϕ(xky)αn (dk) = ϕ(x)ϕ(y).
n→∞
K(n)
50
The function ϕ can be written as
ϕ(g) = π(g)u|u ,
where π is a unitary representation of G on a Hilbert space H, u ∈ HK
and is cyclic. We will show that HK = Cu, and this implies that π is
irreducible. In fact we will show that
P π(g)u = ϕ(g)u.
By assumption
Z
ϕ(xky)αn (dk)
ϕ(x)ϕ(y) = lim
n→∞
K(n)
= lim π(x)Pn π(y)u|u
n→∞
= π(x)P π(y)u|u .
This can be written
P π(y)u|π(x−1 )u = ϕ(y) u|π(x−1 )u ,
and, since u is cyclic, this implies that
P π(y)u = ϕ(y)u.
Examples
1) G(n) = O(n) n Rn is the motion group, K(n) = O(n). Then
G = O(∞) n R(∞) ,
K = O(∞).
A K-biinvariant continuous function ϕ on G can be seen as a radial
function on R(∞) , and can be written
ϕ(x) = Φ(kxk2 ),
where Φ is a continuous function on [0, ∞[, and then, if a = kx · 0k,
b = ky · 0k (x, y ∈ G),
Z
ϕ(xky)αn (dk)
K(n)
Z π
Γ n+1
2 Φ(a2 + b2 + 2ab cos θ) sinn−1 (θ)dθ.
=√
n
πΓ 2 0
51
For every continuous function f on [0, π],
Z π
Γ n+1
π
f (θ) sinn−1 θdθ = f
lim √ 2 n .
n→∞
2
πΓ 2 0
Therefore, if ϕ is a spherical function,
Φ(a2 + b2 ) = Φ(a2 )Φ(b2 ).
It follows that there exists λ ∈ C such that
Φ(u) = e−λu ,
2
ϕ(x) = e−λkxk .
Since ϕ is of positive type, necessarily λ is real, and λ ≥ 0. Hence the
spherical functions of the spherical pair (G, K) are the following
ϕ(x) = e−λkxk
2
(λ ≥ 0).
Is it remarkable that such a function, which is defined on R(∞) , extends
as a continuous function on `2 (N).
2) G(n) = SO(n + 1), K(n) ' SO(n) is the subgroup of the g ∈ G(n)
such g · e0 = e0 , {e0 , . . . , en } being the canonical basis of Rn+1 . A Kbiinvariant continuous function ϕ on G can be written
ϕ(g) = Φ (g · e0 |e0 ) ,
where Φ is a continuous function on [−1, 1]. For such a function, if
(x · e0 |e0 ) = cos a, (y · e0 |e0 ) = cos b (x, y ∈ G),
Z
ϕ(xky)αn (dk)
K(n)
Z π
Γ n+1
Φ(cos a cos b + sin a sin b cos θ) sinn−1 θdθ.
= √ 2 n
πΓ 2 0
If ϕ is spherical, it follows similarly that
Φ(cos a cos b) = Φ(cos a)Φ(cos b).
This implies that there exists an integer m ≥ 0 such that
ϕ(g) = (g · e0 |e0 )m .
52
This function ϕ is of positive type. Therefore the spherical functions
are the following:
ϕ(g) = (g · e0 |e0 )m
(m ∈ N).
3) G(n) = SO0 (1, n), K(n) = SO(n). A K-biinvariant continuous
function on G can be seen as a continuous function on the hyperboloid
with one sheet
x20 − x21 − · · · − x2n = 1, x0 > 0,
and can be written
ϕ(g) = Φ [g · e0 , e0 ] ,
where, for x, y ∈ R(∞) ,
[x, y] = x0 y0 −
∞
X
xn yn ,
n=1
and Φ is a continuous function on [1, ∞[. Furthermore, if [x · e0 , e0 ] =
cosh a, [y · e0 , e0 ] = cosh b (x, y ∈ G),
Z
ϕ(xky)αn (dk)
K(n)
Z π
Γ n+1
Φ(cosh a cosh b + sinh a sinh b cos θ) sinn−1 θdθ.
= √ 2 n
πΓ 2 0
If ϕ is spherical, then
Φ(cosh a cosh b) = Φ(cosh a)Φ(cosh b).
Hence there exists λ such that
ϕ(x) = x−λ
0 .
This function is of positive type if and only if λ is real and ≥ 0. Therefore
the spherical functions are the following
ϕ(x) = x−λ
0
53
(λ ≥ 0).
2. The multiplicative property of the spherical functions.
We assume that G(n), K(n) is one of the examples given below
(a)
- G(n) = O(n) n Sym(n, R), K(n) = O(n).
- G(n) = U (n) n Herm(n, C), K(n) = U (n).
- G(n) = Sp(n) n Herm(n, H), K(n) = Sp(n).
In these three cases G(n)/K(n) = V (n) = Herm(n, F)
(b)
- G(n) = U (n), K(n) = O(n).
- G(n) = U (n) × U (n), K(n) = U (n).
- G(n) = U (2n), K(n) = Sp(n).
In these three cases G(n)/K(n) = Σ(n) is the Shilov boundary of a
Hermitian symmetric space of tube type.
(c)
- G(n) = GL(n, R), K(n) = O(n).
- G(n) = GL(n, C), K(n) = U (n).
- G(n) = GL(n, H), K(n) = Sp(n).
In these three cases G(n)/K(n) = Ω(n) is a symmetric cone.
Let D ⊂ G be the subgroup of diagonal matrices. In case (a) the
entries of a diagonal matrix in G are reals, in case (b) they are in T,
and in case (c) they are positive numbers.
Theorem V.2.1. — Let ϕ ∈ P. The function ϕ is spherical if and
only if there exists a function Φ defined on
R in case (a),
T in case (b),
R+ in case (c),
such that
ϕ diag(a1 , . . . , an ) = Φ(a1 ) . . . Φ(an ).
We will need some preliminaries about the asymptotic behaviour of
the normalized Haar measure αn of K(n).
For n ≥ m let Km (n) be the following subgroup of K(n):
Km (n) =
n
Im
0
0
v
o
v ∈ U (n − m; F) ' U (n − m; F),
and
Km =
∞
[
Km (n) ⊂ K.
n=m
54
Define also
K(m, n) =
n
u
0
0
v
o
u ∈ U (m; F), v ∈ U (n − m; F)
' U (m; F) × U (n − m; F).
The coset space K(n)/K(m, n) is a compact symmetric space, of rank
m if n ≥ 2m. For θ = (θ1 , . . . , θm ) ∈ Rm define


cos θ1
− sin θ1
..
..


.
.




cos θm
− sin θm




cos θ1
a(θ) =  sin θ1
.


..
..


.
.




sin θm
cos θm
In−2m
The set Am of the matrices
a(θ) is a Cartan subgroup for the symmetric
pair K(n), K(m, n) . By the Cartan decomposition, every element k
of K(n) can be written
k = h1 a(θ)h2 ,
where h1 , h2 ∈ K(m, n). Let am = Lie(Am ) ⊂ k(n) = Lie K(n) . It is
the space of the matrices


0
−θ1
..
..


.
.




0
−θm




H(θ) =  θ1
0
.


.
.


..
..




θm
0
0n−2m
For the pair k(n), am , the roots and the multiplicities are the following
√
−1 (±θi ± θj )
√
± −1 (2θi )
√
± −1 θi
(i 6= j)
d
d−1
(n − 2m)d
where d = dimR F (d = 1, 2, 4). The Weyl integration formula
corresponding to the Cartan decomposition is given as
Z
f (k)αn (dk)
K(n)
Z
Z
=
f h1 a(θ)h2 dh1 dh2 Dm,n (θ)dθ1 . . . dθm ,
m
[0, π
2]
K(m,n)×K(m,n)
55
with
Dm,n (θ) = am,n
Y
d
d
sin(θi + θj ) sin(θi − θj )
·
1≤i<j≤m
m
Y
(sin 2θi )d−1 (sin θi )(n−2m)d .
i=1
(See [Helgason,1984], Ch. I, Theorem 5.10.) The constant am,n is
determined by the condition
Z
Dm,n (θ)dθ1 . . . dθm = 1
m
[0, π
2]
Theorem V.2.2. — Let f be a continuous function on K which
is Km -biinvariant, then
Z
Z
lim
f (k)αn (dk) =
f (h1 wm h2 )αm (dh1 )αm (dh2 ),
n→∞
K(n)
K(m)×K(m)
where
0
 Im
=


wm
−Im
0


.

1
..
.
Lemma V.2.3. —
Let X be a compact space, and µ a positive
measure such that every non empty open set has a positive measure.
Let δ ≥ 0 be a continuous function on X which attains its maximum at
only one point x0 . Define
Z
1
=
δ(x)n µ(dx),
an
X
and, for a continuous function f on X,
Z
Ln (f ) = an
f (x)δ(x)n µ(dx).
X
then
lim Ln (f ) = f (x0 ).
n→∞
56
Proof.
For 0 < α < M = max δ, there exists a constant Cα > 0 such that
an ≤ Cα α−n .
In fact there is a neighborhood V of x0 such that δ(x) ≥ α for x ∈ V ,
and
1
≥ µ(V )αn .
an
Let W be a neighborhood of x0 . For x ∈ X \ W , δ(x) ≤ β < M .
Choose α such that β < α < M . Then
Z
an
X\W
β n
δ(x)n µ(dx) ≤ Cα µ(X)
,
α
and
Z
δ(x)n µ(dx) = 0.
lim an
n→∞
X\W
From Lemma V.2.3 it follows that, if f is a continuous function on
[0, π2 ]m , then
Z
lim
n→∞
f (θ)Dm,n (θ)dθ1 . . . dθm = f
m
[0, π
2]
π
2
,...,
π
.
2
Proof of Theorem V.2.2
If f is Km -biinvariant, then
Z
Z
Z
f (k)αn (dk) =
m
[0, π
2]
K(n)
K(m)×K(m)
f h1 a(θ)h2 αm (dh1 )αm (dh2 )Dm,n (θ)dθ1 . . . dθm .
By using Lemma V.2.3, and noticing that
π
π
wn = a , . . . ,
,
2
2
we obtain
Z
lim
n→∞
K(n)
Z
f (k)αn (dk) =
f (h1 wm h2 )αm (dh1 )αm (dh2 ).
K(m)×K(m)
57
Corollary V.2.4. — Let ϕ be a continuous function on G which
is K-biinvariant. For x = diag(a1 , . . . , am ), y = diag(b1 , . . . , bm ),
Z
lim
ϕ(xky)αn (dk) = ϕ diag(a1 , . . . , am , b1 , · · · , bm ) .
n→∞
K(n)
Proof.
The function k 7→ ϕ(xky) is Km -biinvariant. Hence we can apply
Theorem V.2.2 :
Z
ϕ(xky)αn (dk)
lim
n→∞ K(n)
Z
=
ϕ(xh1 wm h2 y)αm (dh1 )αm (dh2 ).
K(m)×K(m)
The statement follows since, for h1 , h2 ∈ K(m),
−1
xh1 wm h2 ywm
∈ Kdiag(a1 , . . . , am , b1 , . . . , bm )K.
Proof of Theorem V.2.1
Let ϕ ∈ P. If ϕ is spherical, then, for x = diag(a1 , . . . , am ),
y = diag(b1 , . . . , bm ),
Z
lim
ϕ(xky)αn (dk) = ϕ(x)ϕ(y).
n→∞
K(n)
By Corollary V.2.4 it follows that
ϕ diag(a1 , . . . , am ) ϕ diag(b1 , . . . , bm )
= ϕ diag(a1 , . . . , am , b1 , . . . , bm ) ,
and that
ϕ diag(a1 , . . . , am ) = Φ(a1 ) . . . Φ(am ),
where Φ is the restriction of ϕ to G(1).
Conversely, if
ϕ diag(a1 , . . . , am ) = Φ(a1 ) . . . Φ(am ),
it follows from Corollary V.2.4 that
Z
lim
ϕ(xky)αn (dk) = ϕ(x)ϕ(y),
n→∞
K(n)
and that ϕ is spherical.
58
Chapter VI
HARMONIC ANALYSIS ON THE SPACE
OF INFINITE DIMENSIONAL HERMITIAN MATRICES
1 Spherical functions and ergodic measures. — Let (X, B) be
a measurable space, and K a group of measurable transformations of
X. Let M be the set of K-invariant probability measures on X. The
set M is convex. Fix µ ∈ M. A measurable set E ⊂ X is said to be
K-invariant relatively to µ if, for every g ∈ K,
µ (gE)∆E = 0.
The measure µ is said to be ergodic if, for every measurable set E which
is K-invariant relatively to µ,
µ(E) = 0 or 1.
Proposition VI.3.1. — Let µ ∈ M. The following properties are
equivalent:
(a) µ is ergodic.
(b) µ is extremal in M.
(c) The subspace of K-invariant functions in L2 (X, µ) reduces to
constant functions.
([Phelps,1966], Proposition 10.4.)
If X is a locally compact topological space, and K a compact group
acting on X by homeomorphisms, then the ergodic measures are exactly
the orbital measures:
Z
µa (f ) =
f (g · x)α(dg) (a ∈ X),
G
where α is the normalized Haar measure of K.
Let us consider the following special case: X = V is a finite dimensional Euclidean space, and K is a closed subgroup of the orthogonal
group O(V ). In that case the ergodic measures are the orbital measures.
Let P be the set of continuous functions of positive type on V which
are K-invariant, with ϕ(0) = 1. The Fourier transform maps bijectively
M onto P, and ext(M) onto ext(P).
59
Let G = K n V be the associated affine motion group. A Kbiinvariant function ϕ on G can be seen as a K-invariant function on
V . If ϕ is continuous of positive type, with ϕ(0) = 1, it is the Fourier
transform of a probability measure µ on V . The function ϕ is spherical
if and only if the measure µ is an orbital measure for the group K.
Assume now that V (n) is an increasing sequence of Euclidean spaces,
V (n) ⊂ V (n + 1), K(n) is a closed subgroup of the orthogonal group
O V (n) , and
K(n) = K(n + 1) ∩ O V (n) .
Define
∞
[
X=V =
V (n),
n=1
equipped with the inductive limit topology,
K=
∞
[
K(n).
n=1
Here X = V ∗ is the dual space of V . It is the projective limit of the
sequence V (n) relatively to the orthogonal projections
pm,n : V (n) → V (m)
(n > m).
(n)
(n)
For each n consider an orthonormal basis {e1 , . . . , edn } of the orthogonal complement of V (n) in V (n + 1). Then
(n)
{ei
| 1 ≤ i ≤ d n , n ∈ N∗ }
is an orthonormal basis of V , hence V can be identified with R(∞)
and V ∗ with R∞ . The group K acts on V ∗ . Let M be the set of
K-invariant probability measures on V ∗ . In this case also the Fourier
transform maps bijectively M onto P, and ext(M) onto ext(P).
Define
G=
∞
[
G(n) = K n V.
n=1
Then (G, K) is a spherical pair. A K-biinvariant continuous function
ϕ on G, which is of positive type, with ϕ(0) = 1, seen as a K-invariant
function on V , is the Fourier transform of a probability measure µ on V ∗ .
The restriction ϕn of ϕ to V (n) is the Fourier transform of a probability
60
measure µn on V (n), and {µn } is a consistent family of measures. In
fact µn = pn (µ), where
pn : V ∗ → V (n)
is the projection of V ∗ onto V (n). The function ϕ is spherical if and
only if the measure µ is ergodic with respect to K. The spherical
representation π of G associated to the spherical function ϕ can be
realized on the Hilbert space H = L2 (V ∗ , µ). For (k, a) ∈ G, f ∈ H,
π(k, a)f (ξ) = e−ha,ξi f (k · ξ).
By using the following formula for the product in G:
(k1 , a1 )(k1 , a2 ) = (k1 k2 , k1 · a2 + a1 ),
one checks that π is a representation. It is clearly unitary. The Kinvariant vectors are the constants, and
Z
π(k, a)1|1) =
e−ha,ξi µ(dξ) = ϕ(a).
V∗
Example 1
In Chapter I we saw the case of
V (n) = Rn , K(n) = O(n),
V = R(∞) , K = O(∞), V ∗ = R∞ .
The spherical functions of positive type for the pair G(n), K(n) are
the Bessel functions:
ϕr (x) = Jn (rkxk)
(r ≥ 0),
and the orbital measures are the uniform spherical measures σr . The
spherical functions for the pair (G, K) are the Gaussian functions
t
2
ϕt (x) = e− 2 kxk ,
and the ergodic measures are the Gaussian measures γt (t > 0), and the
Dirac measure δ at 0.
61
Example 2
The main topic of these notes is the case V (n) = Herm(n, C),
K(n) ' U (n) acting on V (n) by the transformations
k : x 7→ k · x = uxu∗
u ∈ U (n) ,
and
V = H(∞), K ' U (∞).
A spherical function of positive type for the pair G(n), K(n) is the
Fourier transform of an orbital measure. In Chapter II (Theorem
II.4.1) we established a formula for such a function: if µ is the orbital
measure associated to the orbit of a = diag(a1 , . . . , an ), then, for
x = diag(x1 , . . . , xn ),
1
det (e−iaj xk )1≤j,k≤n .
ϕa (x) = δ!
D(a)D(−ix)
The spherical functions of the pair (G, K) are the functions of the
following form:
ϕ diag(x1 , . . . , xn ) = Φ(x1 ) . . . Φ(xn ),
where Φ is a Pólya function. This follows from the theorem of Pickrell
(Theorem IV.4.1), and Theorem V.2.1.
We saw in Section 3 of Chapter I that in Example 1 an ergodic
measure on V ∗ = R∞ with respect to the action of K = O(∞) is the
limit of a sequence of orbital measures on V (n) = Rn for the action of
O(n) (Proposition I.3.1), or, equivalently, that a spherical function ϕ
for the pair (G, K) is the limit of a sequence ϕn , where ϕn is a spherical
function for the pair G(n), K(n) . It turns out that it is a general fact.
In the last section of this chapter we will present a proof of this fact in
the case of Example 2, following closely [Olshanski-Vershik,1996]. We
will need first some preparation about convergence of Pólya functions.
2. Convergence of Pólya functions. — By Theorem IV.4.1,
every Pólya function has a representation as an infinite product
Φ(λ) = Φ(λ; α, β, γ) =
1
2
e−iβλ e− 2 γλ
∞
Y
k=1
with β ∈ R, γ ≥ 0, αk ∈ R,
∞
X
αk2 < ∞.
k=1
62
eiαk λ
,
1 + iαk λ
In the following it will be useful to consider the logarithmic derivative
of the Pólya function Φ:
∞
X
iαk Φ0 (λ)
= −iβ − γλ +
iαk −
Φ(λ)
1 + iαk λ
k=1
∞
X
= −iβ − γ + p2 (α) λ +
pm+1 (−iα)λm ,
m=2
where pm is the m-power sum (see Section 3 of Chapter III), and
|λ| <
1
.
sup |αk |
We consider on the set Ω of Pólya functions the topology of uniform
convergence on compact sets of R. The topological space Ω is metrizable
and complete. In fact if Φ is a limit of Pólya functions, and is continuous,
then Φ is a Pólya function. We would like to express this topology in
terms of the parameters α, β, γ.
First let us observe that, if a sequence Φn of Pólya functions converges
to Φ, then, in general, the corresponding sequence γ (n) does not converge
to γ. In fact, let us consider the following example. For α 6= 0, define
ei √αn λ n
Φn (λ) =
,
1 + i √αn λ
i.e. β (n) = 0, γ (n) = 0, and
(n)
αk
=
√α
n
0
if k ≤ n,
.
otherwise.
Then one checks easily that
1
2
lim Φn (λ) = e− 2 α
n→∞
λ2
= Φ(λ).
Notice that γ (n) = 0, γ = α2 .
To a Pólya function Φ(λ; α, β, γ) we associate a bounded positive
measure σ on R defined by:
σ=
∞
X
αk2 δαk + γδ0 .
k=1
63
If f is a bounded continuous function,
Z
f (t)σ(dt) =
R
∞
X
αk2 f (αk ) + γf (0).
k=1
Let
Theorem VI.2.1. —
Φn (λ) = Φ(λ; α(n) , β (n) , γ (n) ),
Φ(λ) = Φ(λ; α, β, γ),
be Pólya functions, and let σ (n) , σ be the associated measures as above.
Assume that
lim β (n) = β,
n→∞
and, for every bounded continuous function f on R,
Z
Z
(n)
lim
f (t)σ (dt) =
f (t)σ(dt).
n→∞
R
R
Then
lim Φn (λ) = Φ(λ)
n→∞
uniformly on compact sets of R.
Lemma VI.2.2. —
Fourier transform:
Let µ be a probability measure on R, and ϕ its
Z
∞
e−itλ µ(dt).
ϕ(λ) =
−∞
Assume that ϕ admits a power series expansion for |λ| < R:
ϕ(λ) =
∞
X
am λm .
m=0
Then ϕ has a holomorphic extension to the strip
ΣR = {ζ = λ + iη | |η| < R}.
For |η| < R,
Z
∞
etη µ(dt) < ∞,
−∞
and, for ζ ∈ ΣR ,
Z
∞
e−itζ µ(dt).
ϕ(ζ) =
−∞
64
This a special case of a general result by Graczyk and Loeb in [1994]
(Theorem 1).
Proof.
a) We show first that, since ϕ is C ∞ on ] − R, R[, µ has moments of
all orders: for all k,
Z
∞
|t|k µ(dt) < ∞.
−∞
For ε > 0,
2ϕ(0) − ϕ(ε) − ϕ(−ε)
=2
ε2
Z
∞
1 − cos εt
µ(dt).
ε2
−∞
By Fatou’s Lemma,
Z ∞
2ϕ(0) − ϕ(ε) − ϕ(−ε)
t2 µ(dt) ≤ lim
= −ϕ00 (0).
2
ε→0
ε
−∞
It follows that
∞
Z
00
e−itλ t2 µ(dt).
−ϕ (λ) =
−∞
By repeating the argument we obtain, for all `,
Z ∞
t2` µ(dt) < ∞,
−∞
and also
am
1 (m)
(−i)m
=
ϕ (0) =
m!
m!
Z
∞
tm µ(dt).
−∞
b) We show now that, for |η| < R,
Z ∞
cosh(tη)µ(dt) < ∞.
−∞
For |η| < R,
1
2
ϕ(iη) + ϕ(−iη) =
=
=
∞
X
1
2
am (iη)m + (−iη)m
m=0
∞
X
a2` (−1)` η 2`
`=0
∞
X
1
(2`)!
`=0
∞
Z
=
Z
∞
X
−∞ `=0
∞
∞
(tη)2` µ(dt)
−∞
1
(tη)2` µ(dt)
(2`)!
Z
=
cosh(tη)µ(dt).
−∞
65
Lemma VI.2.3. — Let µk be a sequence of probability measures on
R, and ϕk the Fourier transform of µk . One assumes that each function
ϕk has a power series expansion for |λ| < R, hence a holomorphic
extension to the disc
DR = {ζ ∈ C | |ζ| < R}.
Assume moreover that the sequence ϕk converges to a function ϕ
uniformly on the disc Dr for all r < R. Then the sequence µk converges
weakly to a probability measure µ having ϕ as its Fourier transform.
The functions ϕk and ϕ, the Fourier transform of µ, have holomorphic
extensions to the strip
ΣR = {ζ = λ + iη | |η| < R},
and ϕk converges to ϕ uniformly on every compact Q ⊂ ΣR .
In order to apply this Lemma one usually proves that
ϕk (λ) =
∞
X
ak,m λm
(|λ| < R),
m=0
lim ak,m = am ,
k→∞
and
|ak,m | ≤ um ,
with
∞
X
um rm < ∞,
m=0
for r < R.
Proof.
By Lemma VI.2.2, the functions ϕk have holomorphic extensions to
the strip ΣR . For 0 ≤ η < R,
Z ∞
cosh(tη)µk (dt) = 12 ϕk (iη) + ϕk (−iη) .
−∞
Since
1
k→∞ 2
lim
ϕk (iη) + ϕk (−iη) =
1
2
ϕ(iη) + ϕ(−iη) ,
there exists a constant M (η) > 0 such that, for all k,
Z ∞
cosh(tη)µk (dt) ≤ M (η).
−∞
66
For ζ ∈ ΣR ,
Z
∞
e−itζ µk (dt),
ϕk (ζ) =
−∞
and, for ζ ∈ Σr , r < R,
|ϕk (ζ)| ≤ 2M (r).
By the theorem of Montel, a subsequence converges uniformly on every
compact Q ⊂ ΣR . Since ϕk converges on DR , it follows that the
sequence ϕk itself converges uniformly on every compact Q ⊂ ΣR .
Proof of Theorem VI.2.1
Since the sequence
p2 α(n) + γ (n)
is convergent, there exists R > 0 such
p2 α(n) + γ (n) ≤ R2 .
It follows that the measures σ (n) and σ are supported by [−R, R].
Furthermore, for m ≥ 3,
Z
Z
m−2 (n)
(n)
t
σ (dt) =
tm−2 σ(dt) = pm (α),
lim pm α
= lim
n→∞
and
n→∞
R
R
(n) m−2
pm α(n) ≤ sup |αk |
p2 α(n)
m −1
≤ p2 α(n) 2 p2 α(n)
m
= p2 α(n) 2 ≤ Rm .
Therefore
Φ0n (λ)
Φ0 (λ)
lim
=
,
n→∞ Φn (λ)
Φ(λ)
uniformly on {λ ∈ C | |λ| ≤ r} for r < R, so that
lim Φn (λ) = Φ(λ).
n→∞
We obtain the statement of the theorem by applying Lemma VI.2.3.
For R > 0 we define
ΩR = {Φ(.; α, β, γ) ∈ Ω | −Φ00 (0) = p2 (α) + β 2 + γ ≤ R2 }.
67
Proposition VI.2.5. — The set ΩR is compact.
Proof.
a) The set ΩR is relatively compact. In fact, if Φ is the Fourier
transform of a probability measure µ, then
Z
00
−Φ (0) =
t2 µ(dt),
R
and the set of probability measures µ such that
Z
t2 µ(dt) ≤ R2
R
is relatively compact.
b) Let Q ⊂ Ω be compact. There exists R > 0 such that Q ⊂ ΩR .
Observe that a Pólya function does not vanish. Let λ0 > 0. There exists
A > 0 such that, for Φ ∈ Q,
|Φ(λ0 )| ≥ A,
therefore
2
γλ20
|Φ(λ0 )| = e
∞
Y
k=1
and
γλ20 +
∞
X
2
αk2 λ20 ≤ eγλ0
k=1
1
≥ A2 ,
1 + αk2 λ20
∞
Y
(1 + αk2 λ20 ) ≤
k=1
or
γ + p2 (α) ≤
1
A2 λ20
1
,
A2
.
It follows that the set
Q0 = {Φ0 (λ) = eiβλ Φ(λ) = Φ(λ; α, 0, γ) | Φ ∈ Q}
is relatively compact. Since Q and Q0 are relatively compact, the set of
the numbers β is bounded:
|β| ≤ B.
Therefore Q ⊂ ΩR , with
R2 =
1
A2 λ20
68
+ B2.
c) If Φn = Φ(·; α(n) , β (n) , γ (n) ) converges to Φ(·; α, β, γ), then
lim p2 α(n) + γ (n) = p2 (α) + γ,
n→∞
lim pm α(n) = pm (α) for m ≥ 3,
n→∞
lim β (n) = β.
n→∞
By b) there exists R > 0 such that, for all n,
p2 α(n) + γ (n) + β 2 ≤ R2 .
It follows that the functions Φn are holomorphic for |λ| < R, and, by
Lemma VI.2.2, in ΣR . Furthermore, for r < R, there exists a constant
M (r) > 0 such that
|Φn (λ)| ≤ M (r) for λ ∈ Σr .
From the theorem of Montel, it follows that there is a subsequence Φnj
which converges uniformly on compact sets in ΣR . Since the sequence
itself converges to Φ on R, it follows that the sequence Φn converges to Φ
uniformly on compact sets in ΣR . Therefore the logarithmic derivatives
Φ0n
Φn
converge uniformly in a neighborhood of 0, as do the coefficients of their
Taylor expansions at 0.
d) The map
Φ(·; α, β, γ) 7→ p2 (α) + γ,
Ω→R
is continuous. It follows that ΩR is closed.
3. Ergodic measures are limits of orbital measures. — Let µ
be a probability measure on H∞ , which is ergodic with respect to the
action of U (∞). Its Fourier transform ϕ is a spherical function for the
spherical pair (G, K) (see Example 2 in Section 1), and has the form
ϕ diag(x1 , . . . , xn , 0, . . .) = Φ(x1 ) . . . Φ(xn ),
where Φ is a Pólya function. This Pólya function can be written as an
infinite product
Φ(λ) = Φ(λ; α, β, γ) =
1
2
e−iβλ e− 2 γλ
∞
Y
k=1
69
eiαk λ
.
1 + iαk λ
We will see that there exists a sequence µn of orbital measures which
converges to µ: there exists a sequence a(n) of diagonal matrices
(n)
a(n) = diag(a1 , . . . , a(n)
n , 0, . . .),
such that µ is the weak limit of the sequence µn of orbital measures
defined by
Z
Z
f (x)µn (dx) =
f (ua(n) u∗ )αn (du).
U (n)
As we will see this makes possible to relate the asymptotic behaviour of
the sequence a(n) to the parameters α, β, γ of the Pólya function Φ.
To the diagonal matrix a(n) one associates the measure σ (n) on R by
σ
(n)
n
X
1 (n) 2
δ 1 a(n) .
a
=
n k
n2 k
k=1
If f is a bounded continuous function,
Z
f (t)σ
(n)
R
n
X
1 (n) 2 1 (n) a
a
.
(dt) =
f
n2 k
n k
k=1
Here is the fundamental theorem of Olshanski and Vershik in [1996]
(Theorem IV.1), in a slightly different formulation.
Assume that
Theorem VI.3.1. —
1
tr a(n) = β,
n→∞ n
lim
and that
lim σ (n) = σ
n→∞
weakly, with
Z
f (t)σ(dt) =
R
∞
X
αk2 f (αk ) + γf (0).
k=1
Then the measure µn converges weakly to an ergodic measure µ, whose
Fourier transform is given by
ϕ diag(λ1 , . . . , λn ) = Φ(λ1 ) . . . Φ(λn ),
where Φ is the following Pólya function
Φ(λ) =
1
2
e−iβλ e− 2 γλ
∞
Y
k=1
70
eiαk λ
.
1 + iαk λ
Proof.
(a) Let us consider the following sequence of Pólya functions
Φ
(n)
(λ) =
n
Y
1
(n)
1
k=1 1 + i n ak λ
.
The function Φ(n) corresponds to the parameters α(n) , β (n) , γ (n) , with
α(n) =
β (n) =
γ
(n)
1
n
n
1X
n
1 (n)
an , 0, . . . ,
n
(n)
a1 , . . . ,
(n)
ak ,
k=1
= 0.
By Theorem V.2.1, Φ(n) converges to the Pólya function Φ,
1
∞
Y
2
Φ(λ) = e−iβλ e− 2 γλ
k=1
eiαk λ
.
1 + iαk λ
The function Φ(n) has the following power series expansion near 0:
Φ(n) (λ) =
=
∞
X
hm
m=0
∞
X
1
n
a(n) (−iλ)m
cn,m λm ,
m=0
where hm is the complete symmetric function (see the end of Section
II.2).
The function Φ has also a power series expansion:
Φ(λ) =
∞
X
cm λm .
m=0
It follows from the proof of Theorem VI.2.1 that
cm = lim cn,m .
n→∞
Furthermore, there exists R > 0 such that
|cn,m | ≤ Rm .
71
This estimate can be obtained by using the Cauchy inequalities, or the
following lemma:
Lemma VI.3.2. —
Let
f (z) =
∞
X
am z m ,
a0 6= 0.
m=0
Assume that
∞
X
f 0 (z)
=
bm z m ,
f (z)
m=0
with |bm | ≤ Rm+1 . Then |am | ≤ |a0 |Rm .
Proof.
From
Z z
f (z) = f (0) exp
0
f 0 (t) dt ,
f (t)
it follows that
am = a0 Pm (b0 , b1 , . . . , bm−1 ),
where Pm is a polynomial with positive coefficients. If bm = Rm+1 , then
am = a0 Rm . In fact
f 0 (z)
R
=
,
f (z)
1 − Rz
f (z) =
a0
.
1 − Rz
(b) By Theorem II.4.1 the Fourier transform ϕn of µn is given by
X
1
ϕn diag(λ1 , . . . , λn ) = δ!
sm a(n) sm (−iλ).
(m + δ)!
m1 ≥···≥mn ≥0
Since
sm (λ, 0, . . . , 0) =
λm
0
if m = (m, 0, . . . , 0),
otherwise,
and, if m = (m, 0, . . . , 0),
sm (x) = hm (x) =
X
xm ,
|m|=m
it follows that
ϕn diag(λ, 0, . . . , 0)
∞
X
1
= (n − 1)!
hm a(n) (−iλ)m
(m + n − 1)!
m=0
=
∞
X
1
nm
hm a(n) (−iλ)m .
n(n + 1) · · · (n + m − 1)
n
m=0
72
This can be written
ϕn
∞
X
diag(λ, 0, . . . , 0) =
nm
cn,m λm .
n(n + 1) . . . (n + m − 1)
m=0
Since
nm
≤ 1,
n(n + 1) . . . (n + m − 1)
nm
= 1,
n→∞ n(n + 1) . . . (n + m − 1)
lim
it follows from Lemma VI.2.3 that
∞
X
lim ϕn diag(λ, 0, . . . , 0) =
cm λm = Φ(λ).
n→∞
m=0
(c) Let us show now that, for k fixed,
lim ϕn diag(λ1 , . . . , λk , 0 . . .) = Φ(λ1 ) . . . Φ(λk ).
n→∞
If mk+1 > 0, then
sm (λ1 , . . . , λk , 0, . . .) = 0.
Hence
ϕn diag(λ1 , . . . , λk , 0, . . .)
X
1
= δ!
sm a(n) sm (−iλ1 , . . . , −iλk )
(m + δ)!
m1 ≥···≥mk ≥0
X
= δ!
m1 ≥···≥mk ≥0
1
n|m|
(n)
sm a
sm (−iλ1 , . . . , −iλk ).
(m + δ)!
n
On the other hand, by using the identity
k
Y
1
=
1
−
x
y
i
j
i,j=1
X
sm (x)sm (y)
m1 ≥···≥mk ≥0
(Proposition II.3.4), one obtains
Φ
(n)
(λ1 ) . . . Φ
(n)
(λk ) =
X
sm
m1 ≥···≥mk ≥0
73
1
n
(n)
a
sm (−iλ1 , . . . , −iλn ).
By generalizing the proof in (b) one obtains
lim ϕn diag(λ1 , . . . , λk , 0, . . .) = Φ(λ1 ) · · · Φ(λk ).
n→∞
By a multivariate analogue of Lemma VI.2.3 this implies the statement
of Theorem VI.3.1.
4. Is there a Bochner type theorem for spherical pairs ?.
Let (G, K) be a spherical pair (see the definition in Section V.1). We
denoted by P the set of K-biinvariant continuous functions ϕ on G
which are of positive type, with ϕ(0) = 1. Let Ω be the set of spherical
functions of positive type. We saw that the extremal functions in P are
the spherical ones: ext(P) = Ω.
Assume first that (G, K) is a Gelfand pair. For the topology of
uniform convergence on compact sets of G, the set Ω is locally compact.
The Bochner theorem has been extended to this setting by Godement:
Theorem (Bochner-Godement). — If ϕ ∈ P, then there exists
a unique probability measure ν on Ω such that
Z
ϕ(x) =
ω(x)ν(dω).
Ω
In Section V.1, we considered an increasing sequence G(n), K(n)
of Gelfand pairs: G(n) is a closed subgroup of G(n+1), K(n) is a closed
subgroup of K(n + 1), K(n) = K(n + 1) ∩ G(n),
G=
∞
[
G(n),
K=
n=1
∞
[
K(n).
n=1
Then the pair (G, K) is spherical (Theorem V.1.1). It should be natural
to look for an extension of the Bochner-Godement theorem to this
general setting. In several cases the set Ω has been determined and
such a statement has been established.
1. V (n) = Rn , K(n) = O(n), G(n) = O(n) n Rn ,
G = O(∞) n R(∞) ,
K = O(∞).
A K-biinvariant function ϕ on G can be seen as a radial function on
R(∞) . As we saw in Section V.1, the spherical functions are the Gaussian
functions
2
t
ϕt (x) = e− 2 kxk (t ≥ 0).
74
Therefore Ω ' [0, ∞[. And as we saw in Section I.4, every function
ϕ ∈ P can be written
Z
2
t
ϕ(x) =
e− 2 kxk ν(t),
[0,∞[
with a unique probability measure ν on [0, ∞[.
2. G(n) = SO(n+1), K(n) = SO(n). A biinvariant continuous function
ϕ can be written
ϕ(g) = Φ (g · e0 |e0 ) ,
where Φ is a continuous function on [−1, 1]. We saw in Section V.1 that
the spherical functions are the functions
ϕ(g) = (g · e0 |e0 )m
(m ∈ N).
Therefore Ω ' N. Every function ϕ ∈ P can be uniquely written
ϕ(g) = Φ (g · e0 |e0 ) ,
where
Φ(u) =
∞
X
νm um ,
m=0
with
νm ≥ 0,
∞
X
νm = 1.
m=0
([Schoenberg,1942])
3. G(n) = SO0 (1, n), K(n) = SO(n). A K-biinvariant continuous
function ϕ on G can be written
ϕ(g) = Φ [g · e0 , e0 ] ,
where, for x, y ∈ R(∞) ,
[x, y] = x0 y0 −
∞
X
xn yn ,
n=1
and Φ is a continuous function on [1, ∞[. We saw in Section V.1 that
the spherical functions are the functions
ϕ(g) = [g · e0 , e0 ]−λ
75
(λ ≥ 0).
Therefore Ω ' [0, ∞[. Every function ϕ ∈ P can be written
ϕ(g) = Φ [g · e0 , e0 ] ,
where
Z
u−λ ν(λ),
Φ(u) =
[0,∞[
with a unique probability measure on [0, ∞[.
([Krein,1949], [Faraut-Harzallah,1974])
For the examples 1,2,3 and
[Berg-Christensen-Ressel,1984].
related
questions
see
also
4. V (n) = Herm(n, C), K(n) = U (n) acting on V (n) by the
transformations
x 7→ uxu∗
u ∈ U (n) ,
and G(n) = U (n) n Herm(n, C)
We saw that the spherical functions are U (∞)-invariant functions on
H(∞) for which
ϕ diag(a1 , . . . , an ) = Φ(a1 ) . . . Φ(an ),
where Φ is a Pólya function. Therefore ext(P) is parametrized by the
set Ω of Pólya functions. From the work of Borodin and Olshanski
([2001], Theorem 9.1) one can deduce that, for every ϕ ∈ P, there is a
unique probability measure ν on Ω such that
Z
ϕ(x) =
ϕω (x)ν(dω).
Ω
5. Similar results have been obtained about central functions of positive
type on the infinite dimensional unitary group U (∞). Let P be the set
of continuous functions ϕ of positive type on U (∞) which are central:
ϕ(gxg −1 ) = ϕ(x) (g, x ∈ U (∞)), with ϕ(e) = 1. Let us consider the
pair
G = U (∞) × U (∞),
K = diag U (∞) × U (∞) ' U (∞).
It is a spherical pair, and a central function on U (∞) can be seen as a
K-biinvariant function on G.
76
By Theorem V.2.1, if ϕ is a spherical function there is a continuous
function Φ defined on the torus T such that
ϕ diag(t1 , . . . , tn ) = Φ(t1 ) . . . Φ(tn )
(tj ∈ T).
This has been proved in another way by Voiculescu ([1976], Proposition
1). The function Φ is of positive type, and has a Fourier expansion:
Φ(t) =
∞
X
cm tm ,
m=−∞
P∞
with cm ≥ 0, m=−∞ cm = 1. The restriction of ϕ to U (n) has a Schur
expansion (Proposition II.3.1):
ϕ(diag(t1 , . . . , tn ) = Φ(t1 ) . . . Φ(tn ) =
X
a(n)
m sm (t),
m1 ≥···≥mn
where
a(n)
m = det (cmi −i+j )1≤i,j≤n .
The function ϕ is of positive type if and only if, for all n and m,
a(n)
m ≥ 0.
It has been shown by Voiculescu ([1976], Proposition 2) that it holds
for the following functions
−1
Φ(t) = tm eλ(t−1) eµ(t −1)
∞ Y
1 − αk 1 − βk t 1 − γk 1 + δk t−1 ,
1 − αk t
1 − βk
1 − γk t−1
1 + δk
k=1
with
m ∈ Z, λ ≥ 0, µ ≥ 0,
0 ≤ αk < 1, 0 ≤ βk < 1, 0 ≤ γk < 1, 0 ≤ δk < 1,
∞
X
(αk + βk + γk + δk ) < ∞.
k=1
These functions resemble very much to Pólya functions. Later it was
noticed by Vershik and Kerov [1981], and by Boyer [1983], that it holds
if and only if the sequence {cm } is totally positive, i.e., for k1 < · · · < kn ,
`1 < · · · < `n ,
det (cki −`j )1≤i,j≤n ≥ 0.
77
Then, by a theorem of Edrei [1953], the sequence {cm } is totally positive
if and only if the function Φ can be written as an infinite product as
above (as it was conjectured by Schoenberg [1948]).
This parametrization of Ω has been obtained in another way by Vershik and Kerov [1981], by studying the asymptotics of the characters of
U (n) as n → ∞ (see also [Okunkov-Olshanski,1998]). This approach is
more informative since it gives a geometric meaning to this parametrization.
Here also the analogue of the Bochner-Godement theorem holds
([Voiculescu,1976], Théorème 2, see also [Olshanski,2001], Theorem 9.1).
The set Ω of spherical functions (extremal functions in P) is locally
compact (for the topology of uniform convergence on compact sets).
For every function ϕ ∈ P there is a unique probability measure ν on Ω
such that
Z
ϕ(x) =
ω(x)ν(dω).
Ω
6. The infinite symmetric group
S∞ =
∞
[
Sn
n=1
is the group of the bijections g : N∗ → N∗ whose support {k ∈ N∗ |
g(k) 6= k} is finite. For studying central functions on S∞ it is equivalent
to consider K-biinvariant functions for the following spherical pair
G = S∞ × S∞ ,
K = diag(S∞ × S∞ ) ' S∞ ,
The set Ω of spherical functions, or extremal functions in P, is compact
for the pointwise convergence. For every function ϕ ∈ P there is a
unique probability measure ν on Ω such that
Z
ϕ(x) =
ω(x)ν(dx).
Ω
The description of the spherical functions involves functions in one
complex variable with an infinite product representation. Here also
there functions resemble very much to the Pólya functions. An element
g ∈ S∞ is a product of cycles. For m ≥ 2 let γm = γm (g) be the
number of cycles of length m in the decomposition of g. The sequence
{γ2 , γ3 , . . .} determines the conjugacy class of g. If ϕ is spherical, then
it is multiplicative in the following sense:
ϕ(g) =
∞
Y
m=2
78
sγmm (g) ,
where {sm } is a sequence of real numbers, −1 ≤ sm ≤ 1. Let us consider
the generating function of the sequence {sm }:
∞
X
F (z) =
sm+1 z m ,
m=0
and also the Taylor expansion
∞
X
Φ(z) =
cm z m ,
m=0
of the function Φ defined by
Φ0 (z)
= F (z), Φ(0) = 1.
Φ(z)
The function ϕ is of positive type if and only if
δz
Φ(z) = e
∞
Y
1 + βk z
,
1 − αk z
k=1
with
αk ≥ 0, βk ≥ 0,
∞
X
(αk + βk ) ≤ 1.
k=1
Since
∞
∞
k=1
k=1
X βk
X αk
Φ0 (z)
=δ+
+
,
Φ(z)
1 + βk z
1 − αk z
this means that
sm =
∞
X
αkm + (−1)m+1
k=1
∞
X
βkm .
k=1
These results have been proved by Thoma [1964a,1964b]. It turns out
that the function ϕ is of positive type if and only if the sequence {cm } is
totally positive, but Thoma does not refer explicitely to total positivity.
In fact his paper [1964b] contains essentially a proof of Schoenberg’s
theorem about one sided totally positive sequences ([1948]).
These results have been obtained in another way by Vershik and
Kerov ([1981]) by studying asymtotics of the characters of Sn as n → ∞.
79
The symmetric group Sn is the Weyl group of the root system of
type An , and one can say that S∞ is the Weyl group of type A∞ .
Similarly one can consider the infinite Weyl groups of type B∞ , C∞
and D∞ . For these groups the spherical functions (or characters) have
been determined in [Hirai-Hirai,2002].
In [Kerov-Olshanski-Vershik,1993], for z ∈ C, a family Tz of unitary
representations of the spherical pair (G, K) is introduced. The representation Tz decomposes as a direct integral of spherical unitary representations. This decomposition is analysed by studying the integral
representation of the function of positive type
ϕz (g) = Tz (g)f0 |f0 ,
where f0 is a cyclic vector in the representation space.
80
Aissen, M., Schoenberg, I.J., Whitney, A.M. (1952).
On
the generating functions of totally positive sequences I, Journal
d’Analyse Mathématique, 2, 93–103.
Berg, C., Christensen, J. P. R., Ressel, P. (1984). Harmonic
analysis on semigroups. Springer.
Billingsley, P. (1968).
Convergence of probability measures.
Wiley.
Bochner, S. (1959).
Lecture on Fourier integrals.
Princeton
University Press.
Borodin, B., Olshanski, G. (2001). Infinite random matrices and
ergodic measures, Comm. Math. Phys., 223,1, 87–123.
Borodin, S., Olshanski, G. (2002).
Harmonic analysis on the
infinite-dimensional unitary group and determinantal point processes. Preprint.
à paraı̂tre dans Annals of Math.
Boyer, R.P. (1983). Infinite traces of AF-algebras and characters of
U (∞), J. Operator Theory, 9, 205–236.
Edrei, E. (1952).
On the generating functions of totally positive
sequences II, Journal d’Analyse Mathématique, 2, 104–109.
Edrei, E. (1953).
On the generating function of a doubly infinite
totally positive sequence, Trans. Amer. Math. Soc., 74, 367–383.
Faraut, J. (1982). Analyse sur les paires de Gelfand et les espaces
hyperboliques in Analyse harmonique. Les cours du CIMPA.
Faraut, J., Harzallah, K. (1974). Distances hilbertiennes invariantes sur un espace homogène, Ann. Inst. Fourier, 24, 171–217.
Faraut, J. Korányi, A. (1994).
Analysis on symmetric cones.
Clarendon Press, Oxford.
Gangolli, R., Varadarajan, V.S. (1988). Harmonic analysis of
spherical functions on real reductive groups. Springer Verlag.
Graczyk, P., Loeb, J.-J. (1994). Bochner and Schoenberg theorems
on symmetric spaces in the complex case, Bull. Soc. Math. France,
122, 571–590.
Gross, K.I., Richards, D. St. P. (1989). Total positivity, spherical
series, and hypergeometric functions of matrix arguments, J. of
Approximation Theory, 59, 224–246.
Hamburger, H. (1920).
Bemerkungen zu einer Fragestellung des
Herrn Pólya, Math. Zeit., 7, 302–322.
Helgason, S. (1962). Differential geometry and symmetric spaces.
Academic Press.
Helgason, S. (1984). Groups and geometric analysis. Academic
Press.
81
Hirai, T., Hirai, E. (2002).
Characters for the infinite Weyl
groups of type B∞ /C∞ and D∞ . in Non-Commutativity, InfiniteDimensionality, and Probability at the Crossroad (Eds. Obata,
Matsui & Hora), World Scientific Publishing.
Hua, L.K. (1963). Harmonic analysis of functions of several variables
in the classical domains. Amer. Math. Soc.
Jensen, J.L.W. (1912-13). Recherches sur la théorie des équations,
Acta Math., 36, 181–195.
Karlin, S. (1968). Total positivity, vol. I. Stanford Univ. Press,
Stanford, CA.
Kerov, S., Olshanski, G., Vershik, A. (1993). Harmonic analysis
on the infinite symmetric group. A deformation of the regular
representation, C. R. Acad. Sci. Paris, 316, 773–778.
Krein, M.G. (1949).
Hermitian kernels on homogeneous spaces.
Amer. Math. Translations (2), vol. 34, I. 1949, II. 1950.
Krein, M.G., Milman, D.P. (1940). On extreme points of regular
convex sets, Studia Math., 9, 133-138.
Laguerre, E. (1898). Sur les fonctions du genre zéro et du genre un,
Oeuvres de Laguerre I, Paris, , 174–177.
Neeb, K.-H., Ørsted, B. (2001). Representations in L2 -spaces on
infinite-dimensional symmetric cones, J. funct.Anal., 190, 133–178.
Neretin, Y.A. (2002). Hua-type integrals over unitary groups and
over projective limits of unitary groups, Duke Math. J., 114, 239–
266.
Nessonov, N.I. (1983).
Description of representations of a group
of invertible operators of a Hilbert space that contain the identity
representation of the unitary subgroup, Functional Anal. Appl., 17,
64–66.
Nessonov, N.I. (1986). A complete classification of the representations of GL(∞) containing the identity representation of the unitary
subgroup, Math. USSR Sbornik, 58,No 1, 127–147.
Okunkov, A., Olshanski, G. (1998). Asymptotics of Jack polynomials as the number of variables goes to infinity, Intern. Math. Res.
Notices, 13, 641–682.
Olshanski, G. (1978).
Unitary representations of the infinite
dimensional classical groups U (p, ∞), SO(p, ∞), Sp(p, ∞) and the
corresponding motion groups, Soviet Math. Doklady, 19, 220–224.
Olshanski, G. (1978).
Unitary representations of the infinitedimensional classical groups U (p, ∞), SO0 (p, ∞), Sp(p, ∞) and
the corresponding motion groups, Functionnal analysis and its
Applications, 12, 185–195.
82
Olshanski, G. (1984).
Infinite dimensional classical groups of
finite R-rank: description of representations and asymptotic theory,
Funct. Anal. Appl., 18, 22–34.
Olshanski, G. (1987).
Unitary representations of the group
SO(∞, ∞) as limits of unitary representations of the groups
SO(n, ∞) as n → ∞, Funct. Anal. Appl., 20, 292–301.
Olshanski, G. (1989).
Unitary representations of (G, K)-pairs
connected with the infinite symmetric group S(∞), Leningrad Math.
J., 1, 983–1014.
Olshanski, G. (1990). Unitary representations of infinite dimensional
pairs (G, K) and the formalism of R. Howe, in Representations of
Lie groups and related topics (Eds. A.M. Vershik, D.P. Zhelobenko).
Advanced Studies in Contemporary Mathematics, vol. 7. Gordon
and Breach.
Olshanski, G., Vershik, A. (1996).
Ergodic unitarily invariant
measures on the space of infinite Hermitian matrices, Contemporary
Mathematical Physics (R.L. Dobroshin, R.A. Minlos, M.A. Shubin,
A.M. Vershik), Amer. Math. Soc. Translations (2), 175, 137–175.
Olshanski, G (2001). The problem of harmonic analysis on infinitedimensional unitary group. Preprint.
à paraı̂tre dans J. Funct. Anal.
Parthasarathy, K. (1967). Probability measures on metric spaces.
Academic Press.
Pickrell, D. (1987).
Measures on infinite dimensional Grassman
manifolds, J. of Funct. Anal., 70, 323–356.
Pickrell, D. (1990). Separable representations for automorphism
groups of infinite symmetric spaces, J. Funct. Anal., 90, 1–26.
Pickrell, D. (1991).
Mackey analysis of infinite classical motion
groups, Pacific J. Math., 150, 139–166.
Pickrell, D. (2000). Invariant measures for unitary groups associated to Kac-Moody Lie algebras. Mem. Amer. Soc. 146.
Phelps, R.R. (1966). Lecture on Choquet’s theorem. Van Nostrand.
Pólya, G. (1913). Ueber Annaeherung durch Polynome mit lauter
reelen Wurzeln, Rendiconti di Palermo, 36, 1–17.
Pólya, G., Schur, J. (1914). Über zwei Arten von Faktorenfolgen
in der Theorie der albraischen Gleichungen, J. Reine Angewandte
Math., 144, 89–115.
Pólya, G., Szegö, G. (1976). Problems and Theorems in Analysis.
Springer-Verlag.
Richards, D. St. P. (1990). Total positive kernels, Pólya frequency
functions, and the generalized hypergeometric series, Lin. Alg. and
Appl., 137/138, 467–478.
83
Schoenberg, I.J. (1938a). Metric spaces and completely monotone
functions, Ann. of Math., 39, 811–841.
Schoenberg, I.J. (1938b).
Metric spaces and positive definite
functions, Trans. A.M.S., 44, 522–536.
Schoenberg, I.J. (1942).
Positive definite functions on spheres,
Duke Math. J., 9, 96–108.
Schoenberg, I.J. (1948). Some analytical aspects of the problem of
smoothing. Courant Anniversary Volume, New-York, 546–564.
Schoenberg, I.J. (1950).
On Pólya frequency functions. II.
Variation diminishing integral operators of the convolution type,
Acta Scientiarum Mathematicarum, Szeged, 12, 97–106.
Schoenberg, I.J. (1951).
On Pólya frequency functions. I. The
totally positive functions and their Laplace transforms, Journal
d’Analyse Mathématique, 1, 331–374.
Schur, J. (1914).
Zwei Sätze über algebraische Gleichungen mit
lauter reellen Wurzeln, J. reine angewandte Math., 144, 75–88.
Schwartz, L. (1973).
Radon measures on arbitrary topological
spaces and cylindrical measures. Oxford University Press and Tata
Institute of Fundamental Research.
Thoma, E. (1964a).
Über positiv definite Klassenfunktionen
abzählbarer Gruppen, Math. Z., 84, 389–402.
Thoma, E. (1964). Die unzerlegbaren, positiv-definiten Klassen funktionen der abzählbar unendlichen, symmetrischen Gruppe, Math.
Z., 85, 40–61.
Vershik, A., Kerov, S. (1981). Asymptotic theory of characters of
a symmetric group, Funct. Anal. Appl., 15, 246–245.
Vershik, A., Kerov, S. (1982).
Characters and factor representations of the infinite unitary group, Soviet Math. Doklady, 26,
No.3, 570–574.
Voiculescu, D. (1976). Représentations factorielles de type II1 de
U (∞), J. Math. pures et appl., 55, 1–20.
Widder, D.V. (1946). The Laplace transform. Princeton University
Press.
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