ANALYSIS ON HOMOGENEOUS CONES AND
ASSOCIATED TUBE DOMAINS
DAVID BÉKOLLÉ AND CYRILLE NANA
Outline of the Course
Tube domains over homogeneous cones are classical homogeneous domains, that is, domains on which the linear automorphisms act transitively.
Apart from the real half-line, examples of homogeneous cones are given by
the forward light cone (or Lorentz group) or the cone of positive definite
matrices in the space of symmetric matrices. These two examples happen
to have the stronger property of being symmetric. We will see that there
are homogenous cones that are non symmetric, and explain how they can
be described in algebraic terms. Starting from the case of the upper-half
plane, we will progressively see how to find formulas that allow to describe
the holomorphic functions that are square integrable on the associated tube
domains in terms of Laplace transforms. We will then consider continuity
properties of the Bergman projection.
This course will be organized as follows:
(1) Convex Cones, Homogeneous Cones and associated tube domains.
The case of the upper half-plane of C.
(2) Bergman Spaces, Bergman kernels, Bergman Projections in the unit
disc of C and the upper half plane of C.
(3) Lp -Boundedness of the Bergman Projections in some examples of
tube domains over Homogeneous Cones. The case of the Lorentz
cone and the Vinberg’s Cone.
(4) Open problems.
References
[1] Békollé D., A. Bonami, G. Garrigós, C. Nana, M.M. Peloso and F. Ricci, ”Lecture
Notes on Bergman Projectors in tube domains over cones: an analytic and geometric viewpoint”. IMHOTEP J. Afr. Math. Pures Appl. Vol. 5, No 1 (2004). website:
www.univ-orleans.fr/mapmo/imhotep/index.php?
[2] Békollé D., A. Bonami, M.M. Peloso and F. Ricci, ”Boundedness of weighted Bergman
projections on tube domains over light cones”, Math. Zeit. 237 (2001), 31-59.
[3] Békollé D. and C. Nana, ” Lp -boundedness of Bergman projections in the tube domain
over Vinberg’s cone”, Journal of Lie Theory Vol. 17 (2007), No 1, 115-144.
[4] Faraut J. and A. Korányi, Analysis on Symmetric cones, Clarendon Press, Oxford,
(1994).
[5] Gindikin S.G., “Analysis on homogeneous domains”, Russian Math. Surveys 19 (1964),
1-83.
[6] Vinberg E. B., “The theory of convex homogeneous cones”, Trudy Moskov. Mat. Obsc.,
12 (1963), 359-388.
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Overview of the Course
Uncertainty Principles and Prolate Spheroidal Wave
Functions.
Aline Bonami and Abderrazek Karoui
Heisenberg uncertainty principle asserts that a nonzero L2 −function and its Fourier transform
cannot be both localized simultaneously. This principle has an important theoretical value as well as
many real life applications, in a wide range of scientific areas, such as quantum physics or signal processing. In the first part of this course, we will give different formulations of Heisenberg Uncertainty
Principle, starting from the classical Heisenberg inequality. We will mention its link with sampling
and interpolation. This will lead us to the less-known uncertainty inequality of Slepian-Pollack, and
to the special functions that are known as Prolate Spheroidal Wave Functions.
Prolate spheroidal wave functions (PSWFs) are the eigenfunctions of the differential operator
dψ
d2 ψ
− c2 x2 ψ, where c > 0 is a constant.
Lc defined on C 2 ([−1, 1]) by Lc (ψ) = (1 − x2 ) 2 − 2x
dx
dx
Starting from their non-classical version of the uncertainty principle, D. Slepian, H. Pollak and
H. Landau, have discovered that the PSWFs are also the eigenfunctions of a self-adjoint compact
integral operator Fc defined on L2 ([−1, 1]) by
Z
1
Fc (ψ)(x) =
−1
sin c(x − y)
ψ(y) dy.
π(x − y)
(1)
The PSWFs have the desirable and surprising property to form an orthogonal basis of L2 ([−1, 1]), an
orthonormal system of L2 (R) and more importantly,
an orthonormal basis ofoBc , the Paley-Wiener
n
2
space of band-limited functions given by Bc = f ∈ L (R), Suppt fb ⊂ [−c, c] . We should mention
that no other known orthogonal system of L2 (R) does possess this strange property. In the second
part of this course, we study different properties of the PSWFs. Moreover, we describe some efficient
methods for their accurate computation. We show that the set B = {ψn,c (x), n ≥ 0} is well adapted
for the representation of band-limited functions with band-width c > 0. More importantly, we show
that if a function f belongs to the space of almost time-limited and almost band-limited functions,
then f is efficiently approximated by the PSWFs. Finally, we give an extension of the theory of the
PSWFs to the multidimensional case.
References
[1] A. Bonami and B. Demange A survey on uncertainty principles related to quadratic forms.
Collect. Math. 2006 Vol. Extra, 1–36.
[2] J. A. Hogan and J. D. Lakey, Time-frequency and time-scale methods. Adaptive decompositions,
uncertainty principles, and sampling, Birkhäuser Boston, Inc., 2005.
[3] A. Karoui, Uncertainty Principles, Prolate Spheroidal Wave Functions and Applications, to
appear in the Conference Proceedings book, Fractals and Related Fields, Birkhauser-Boston.
[4] A. Karoui and T. Moumni, New efficient methods of computing the prolate spheroidal wave functions and their corresponding eigenvalues, to appear in Applied Computational Harmonic Analysis,
24 (2008), 269–289.
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[5] A. Karoui and T. Moumni, Spectral Analysis of the Finite Hankel Transform and Circular Prolate
Spheroidal Wave Functions, Journal of Computational and Applied Mathematics, 233, No.3 (2009),
315–333.
[6] H. J. Landau and H. O. Pollak, Prolate spheroidal wave functions, Fourier analysis and uncertaintyII, Bell System Tech. J. 40, (1961), 65–84.
[7] D. Slepian, H. O. Pollak, Prolate spheroidal wave functions, Fourier analysis, and uncertainty-I,
Bell System Technical Journal 40 (1961) 43–63.
[8] D. Slepian, Some Comments on Fourier Analysis, Uncertainty and Modeling, SIAM Review, Vol.
25, No.3, 1983.
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Overview of the Course
Best analytic and meromorphic approximation in Hardy classes;
application to inverse potential problems.
Juliette Leblond
Here is a sketch of the questions that will be considered.
• Prerequisites: basic properties of Hardy spaces H p (D), manely for p = 2, ∞ (from Brett’s
part 1); duality relations, extremal problems on T, [8, 10]; Toeplitz and Hankel operators,
Nehari and Adamjan-Arov-Krein theorems [11] (from Sandra).
• Bounded extremal problems in H p (D) on subsets of T, [4, 5, 6]: if I ⊂ T, approximate a
given function in Lp (I) by a function in H p (D) subject to some norm constraint on T \ I.
Existence, uniqueness and constructive issues. Also for annular domains, and in meromorphic
or rational classes of functions.
• Applications: harmonic identification (frequency domain identification for linear stationary
causal controlled systems, transfer functions), inverse problems for Laplace equations in 2D.
• Hardy spaces Hνp (D) of solutions to (conjugate) Beltrami equation, approximation issues
[1, 2, 7, 9]; applications: inverse problems for the conductivity equation in 2D (thermonuclear
fusion, plasma in tokamaks).
• Hardy spaces H p (B) of analytic functions in B ⊂ R3 , approximation issues [3, 12], also
for spherical domains; applications: inverse problems for Laplace equations in 3D (medical
engineering, EEG-MEG).
References
[1] K. Astala, L. Päivärinta, Calderón’s inverse conductivity problem in the plane, Ann. of Math.
(2) 163, no. 1, 265–299, 2006.
[2] K. Astala, L. Päivärinta, A boundary integral equation for Calderón’s inverse conductivity
problem, Proc. 7th Int. Conf. on Harmonic Analysis and PDEs., Madrid (Spain), 2004, Collect. Math. , Vol. Extra, 127–139, 2006.
[3] B. Atfeh, L. Baratchart, J. Leblond, J. R. Partington, Bounded extremal and Cauchy–Laplace
problems on the sphere and shell, Submitted for publication, 2008.
[4] L. Baratchart, J. Leblond, Hardy approximation to Lp functions on subsets of the circle with
1 ≤ p < ∞, Constr. Approx. 14, no. 1, 41–56, 1998.
[5] L. Baratchart, J. Leblond, J. R. Partington, Hardy approximation to L∞ functions on subsets
of the circle, Constr. Approx. 12 , no. 3, 423–435, 1996.
[6] L. Baratchart, J. Leblond, J. R. Partington, Problems of Adamjan–Arov–Krein type on subsets
of the circle and minimal norm extensions, Constr. Approx. 16 (2000), no. 3, 333–357.
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[7] L. Baratchart, J. Leblond, S. Rigat, E. Russ, Hardy spaces of the conjugate Beltrami equation,
http://fr.arxiv.org/abs/0907.0744, Submitted for publication, 2009.
[8] P. L. Duren, Theory of H p spaces, Pure and Applied Mathematics, Vol. 38, Academic Press,
New York–London, 1970.
[9] Y. Fischer, J. Leblond, J. R. Partington, E. Sincich, Bounded extremal problems in Hardy
spaces for the conjugate Beltrami equation in simply-connected domains, In preparation.
[10] J. B. Garnett, Bounded analytic functions, Pure and Applied Math. 96, Academic Press, New
York, 1981.
[11] N. K. Nikolski, Operators, functions, and systems: an easy reading. Vol. 1, Mathematical
Surveys and Monographs, American Mathematical Society, Providence, RI, 2002.
[12] E. M. Stein, G. Weiss, Introduction to Fourier analysis on euclidean spaces, Princeton Univ.
Press, 1971.
2
Overview of the Course
Hankel and Toeplitz Operators and their application in
Control Theory.
Sandra Pott
During the lectures, the following topics will be covered.
1. the Hardy and Bergman spaces in the right half plane, [1, 4, 6, 7] (to coordinate with
Juliette’s, Brett’s, David and Cyrille’s courses)
2. the Laplace transform on L2 ((0, ∞), tα dt) as an isometric isomorphism into the appropriate Hardy- or Bergman space
3. Definition and basic properties of Hankel and Toeplitz operators on the Hardy space
[7]
4. Nehari’s Theorem and the Adamjan-Arov-Krein Theorem [6, 7]
5. The Carleson-Duren Embedding Theorem [1, 5]
6. Admissibility and controllability in linear systems [4]
7. The Weiss conjecture [2, 3, 9]
There are numerous exercises that will be provided during the course so that the interested
student can learn the basics of the topics covered. These exercises will be designed so that
an interested student can go from a limited background in the area to a relatively deep
understanding of the material.
References
[1] P. Duren, Theory of H p spaces, Pure and Applied Mathematics, vol. 38, Academic Press, New YorkLondon, 1970.
[2] B. Jacob and J. R. Partington, The Weiss conjecture on admissibility of observation operators for
contraction semigroups, Int. Eq. Op. Th. 40 (2001), 231-243.
[3] B. Jacob, J. R. Partington, and S. Pott, Admissible and weakly admissible operators for the right shift
semigroup, Proc. Edin. Math. Soc. 45 (2002), 353-362.
[4] J. R. Partington, Linear operators and linear systems, London Mathematical Society Student Texts,
vol. 60, Cambridge University Press, Cambridge, 2004.
[5] L. Carleson, Interpolations by bounded analytic functions and the corona problem, Ann. of Math. (2) 76
(1962), 547–559.
[6] J. B. Garnett, Bounded analytic functions, Pure and Applied Mathematics, vol. 96, Academic Press Inc.
[Harcourt Brace Jovanovich Publishers], New York, 1981.
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Overview of the Course
The Corona Problem.
Brett Wick
The course will focus on certain aspects of the Corona problem for multiplier algebras of
Besov–Sobolev spaces. The Besov–Sobolev spaces of functions are a class of analytic functions that measure the smoothness of the function. The Besov–Sobolev spaces Bσ2 (Bn ) of
analytic functions on the unit ball Bn in Cn are the collection of functions that are analytic
on the unit ball and such that for any integer m ≥ 0 and any 0 ≤ σ < ∞ such that m+σ > n2
we have the following norm being finite:
kf k2B2σ (Bn )
:=
m−1
X
Z
|f
(j)
2
|(1 − |z|2 )m+σ f (m) (z)|2
(0)| +
Bn
j=0
dV (z)
.
(1 − |z|2 )n+1
One can show that these spaces are independent of m and are reproducing kernel Hilbert
spaces, with obvious inner products. Moreover, there are natural generalizations of this norm
to the scale 1 < p < ∞, [16].
We then will discuss the multiplier algebras of the Besov–Sobolev spaces. For the Besov–
Sobolev space Bσ2 (Bn ), one defines the multiplier algebra M2σ (Bn ) as the collection of analytic
functions ϕ that are pointwise multipliers of Bσ2 (Bn ). Namely, ϕf ∈ Bσ2 (Bn ) for all f ∈
Bσ2 (Bn ), and then norms M2σ (Bn ) by
kϕkM2σ (Bn ) :=
kϕf kBσ2 (Bn )
.
f ∈Bσ2 (Bn ) kf kBσ2 (Bn )
sup
These spaces of functions contain all the well-known and studied examples of analytic functions, including the Dirichlet space, the Hardy space, and the Bergman space. For a certain
range of values of σ, these spaces of functions have deep connections with operator theory.
When 0 ≤ σ ≤ 12 the space of function the space Bσ2 (Bn ) possess additional properties, and
has numerous connections with interpolation theory and other problems in complex function
theory, [1, 2].
The ultimate goal of the course will be to discuss the background and ideas in the proof
of the Corona theorem for the multiplier algebras M2σ (Bn ).
Theorem 0.1 (Costea, Sawyer, Wick [9]) Let 0 ≤ σ ≤ 21 . Suppose that g1 , . . . , gN ∈
M2σ (Bn ) satisfy
N
X
0<δ≤
|gj (z)|2 ≤ 1 ∀z ∈ Bn .
j=1
Are there f1 , . . . , fN ∈ X2σ (Bn ) such that
PN
(i)
∀z ∈ Bn ;
j=1 fj (z)gj (z) = 1
1
(ii)
PN
j=1
kfj kX2σ (Dn ) ≤ Cn,N,σ,δ ?
This theorem encompasses many extensions of L. Carleson’s famous proof for H ∞ (D), [7, 8].
During the lectures, the following topics will be covered.
1. In the preliminary lectures, we will focus on the basic aspects of the Besov–Sobolev
spaces of analytic functions and their multiplier algebras, topics will include the Carleson measures and their geometric characterization. In particular, the well known
examples of the Hardy space H 2 (D) and H ∞ (D) will be highlighted and contrasted
with the other spaces of functions. Topics from [3–5, 10, 12, 16] will be covered.
2. Illustrate the connections with operator theory and function theory when 0 ≤ σ ≤ 12 .
In particular some aspects of [1, 6, 11] will be discussed.
3. Discuss the tools of the proof of the Corona problem; topics to include are the ∂problem on Bn , Charpentier solution operators for these equations, and estimates for
these operators. Topics will include results from [9] and the references therein.
There are numerous exercises that will be provided during the course so that the interested
student can learn the basics of the topics covered. These exercises will be designed so that
an interested student can go from a limited background in the area to a relatively deep
understanding of the material. Additionally, open problems and future directions of research
will be pointed out and highlighted during the course.
References
[1] J. Agler and J. E. McCarthy, Pick interpolation and Hilbert function spaces, Graduate Studies in Mathematics, vol. 44, American Mathematical Society, Providence, RI, 2002.
[2]
, Complete Nevanlinna-Pick kernels, J. Funct. Anal. 175 (2000), no. 1, 111–124.
[3] N. Arcozzi, R. Rochberg, and E. Sawyer, Carleson measures for the Drury-Arveson Hardy space and
other Besov-Sobolev spaces on complex balls, Adv. Math. 218 (2008), no. 4, 1107–1180.
[4]
, Carleson measures and interpolating sequences for Besov spaces on complex balls, Mem. Amer.
Math. Soc. 182 (2006), no. 859, vi+163.
[5]
, Carleson measures for analytic Besov spaces, Rev. Mat. Iberoamericana 18 (2002), no. 2, 443–
510.
[6] J. A. Ball, T. T. Trent, and V. Vinnikov, Interpolation and commutant lifting for multipliers on reproducing kernel Hilbert spaces, Operator theory and analysis (Amsterdam, 1997), Oper. Theory Adv.
Appl., vol. 122, Birkhäuser, Basel, 2001, pp. 89–138.
[7] L. Carleson, Interpolations by bounded analytic functions and the Corona problem, Proc. Internat. Congr.
Mathematicians (Stockholm, 1962), 1963, pp. 314–316.
[8]
, Interpolations by bounded analytic functions and the corona problem, Ann. of Math. (2) 76
(1962), 547–559.
[9] Ş. Costea, E. Sawyer, and B. D. Wick, The Corona Theorem for the Drury-Arveson Hardy space
and other holomorphic Besov Sobolev spaces on the unit ball in Cn , submitted to Acta Math.,
http://arxiv.org/abs/0811.0627http://arxiv.org/abs/0811.0627.
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[10] J. B. Garnett, Bounded analytic functions, Pure and Applied Mathematics, vol. 96, Academic Press Inc.
[Harcourt Brace Jovanovich Publishers], New York, 1981.
[11] S. McCullough and T. T. Trent, Invariant subspaces and Nevanlinna-Pick kernels, J. Funct. Anal. 178
(2000), no. 1, 226–249.
[12] N. K. Nikolski, Operators, functions, and systems: an easy reading. Vol. 1, Mathematical Surveys
and Monographs, vol. 92, American Mathematical Society, Providence, RI, 2002. Hardy, Hankel, and
Toeplitz; Translated from the French by Andreas Hartmann.
[13] E. T. Sawyer, Function theory: interpolation and corona problems, Fields Institute Monographs, vol. 25,
American Mathematical Society, Providence, RI, 2009.
[14] K. Seip, Interpolation and sampling in spaces of analytic functions, University Lecture Series, vol. 33,
American Mathematical Society, Providence, RI, 2004.
[15] E. Tchoundja, Carleson measures for the generalized Bergman spaces via a T (1)-type theorem, Ark.
Mat. 46 (2008), no. 2, 377–406.
[16] K. Zhu, Spaces of holomorphic functions in the unit ball, Graduate Texts in Mathematics, vol. 226,
Springer-Verlag, New York, 2005.
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