Helsinki University of Technology Institute of Mathematics Research Reports
Teknillisen korkeakoulun matematiikan laitoksen tutkimusraporttisarja
Espoo 2003
A464
SAMPLING AT EQUIANGULAR GRIDS ON THE 2-SPHERE AND
ESTIMATES FOR SOBOLEV SPACE INTERPOLATION
Ville Turunen
AB
TEKNILLINEN KORKEAKOULU
TEKNISKA HÖGSKOLAN
HELSINKI UNIVERSITY OF TECHNOLOGY
TECHNISCHE UNIVERSITÄT HELSINKI
UNIVERSITE DE TECHNOLOGIE D’HELSINKI
Helsinki University of Technology Institute of Mathematics Research Reports
Teknillisen korkeakoulun matematiikan laitoksen tutkimusraporttisarja
Espoo 2003
A464
SAMPLING AT EQUIANGULAR GRIDS ON THE 2-SPHERE AND
ESTIMATES FOR SOBOLEV SPACE INTERPOLATION
Ville Turunen
Helsinki University of Technology
Department of Engineering Physics and Mathematics
Institute of Mathematics
Ville Turunen: Sampling at equiangular grids on the 2-sphere and estimates for
Sobolev space interpolation; Helsinki University of Technology Institute of Mathematics Research Reports A464 (2003).
Abstract: We sample functions at 4n2 equiangularly spaced points on the
unit sphere S2 of the 3-dimensional Euclidean space, and study the corresponding interpolation projections Qn : C(S2 ) → C(S2 ) in the scale of L2 -type
Sobolev spaces H s (S2 ). The main result is that if −1 < s < t and t > 7/2
then
kf − Qn f kH s (S2 ) ≤ cs,t,ε ns−t n4+ε kf kH t (S2 ) ,
where cs,t,ε < ∞ is a constant depending on s, t and ε > 0. Hence this
result is useful only when t > s + 4. We also compare our methods to the
well-known interpolation on the circle S1 .
AMS subject classifications: 65T40, 33C55, 46E35, 42C10, 65T50.
Keywords: spherical harmonics, sampling, equiangular grid, interpolation, Sobolev space estimates.
[email protected]
ISBN 951-22-6828-0
ISSN 0784-3143
Helsinki University of Technology
Department of Engineering Physics and Mathematics
Institute of Mathematics
P.O. Box 1100, 02015 HUT, Finland
email:[email protected]
http://www.math.hut.fi/
1
Introduction
Let S2 be the unit sphere of the Euclidean space R3 . In this paper, we are
concerned with approximation in the L2 -type Sobolev spaces H s (S2 ). A most
natural set of sampling points on the sphere consists of the intersections of
the equiangularly spaced longitudes and latitudes, and we shall restrict to
this case. We study approximation in Sobolev spaces on the sphere S 2 with
comparison to the very basic case of interpolation on the circle S 1 . For S1 ,
the old results are optimal and easy to obtain. However, non-commutative
harmonic analysis poses hard problems, and for S2 , our results are quite likely
far from optimal, and yet require some work.
Our interest in this subject came originally from a need to solve integral and pseudodifferential equations by efficient algorithms. For example, a
Dirichlet boundary value problem of a linear elliptic partial differential equation in a domain diffeomorphic to the unit disk of the plane R 2 leads to a
pseudodifferential equation on the unit circle S1 ⊂ R2 . To solve such an
equation numerically, one may use equispaced sampling points on S 1 , and
compute with the now-classical FFT, the Fast Fourier Transform (see [8]).
Within last few years, the progress in Fast Fourier type algorithms on the
sphere S2 (see [1], [2], [3]) has made it feasible to tackle applications, ranging
from tomography to meteorology. E.g. boundary value problems in domains
diffeomorphic to the unit ball of R3 can be considered ([6], [7]).
The structure of the paper is as follows: The second section reviews the
well-known analysis on the circle. The third section is the core, dealing with
the case of the sphere. The fourth section demonstrates the sins committed
in estimates, applying the techniques of the sphere case to the circle.
3
2
Approximation on S1
To get some perspective to numerical harmonic analysis on the sphere S 2 ,
let us review some elementary facts from numerical Fourier analysis on the
unit circle S1 ⊂ R2 . We identify S1 naturally with the 1-dimensional torus
T = R/Z (the space of real numbers modulo integers): if x ∈ [0, 1[ then
x + Z ∈ T can be identified with (cos(2πx), sin(2πx)). Of course, we may
identify T also with the interval [0, 1[, via (x + Z 7→ x) : T → [0, 1[. The
Fourier transform fb : Z → C of a measurable function f : S1 → C is defined
by
Z
fb(ξ) =
1
f (x) e−i2πx·ξ dξ.
0
The L2 -type Sobolev space of order s ∈ R on S1 , denoted by H s (S1 ), is the
completion of C ∞ (S1 ) with respect to the norm
!
Ã
¯2 1/2
¯
X
¯
¯
,
hξi2s ¯fb(ξ)¯
f 7→ kf kH s (S1 ) :=
ξ∈Z
where hξi := max{1, |ξ|}; the inner product is of course
X
hξi2s fb(ξ) gb(ξ).
(f, g) 7→ hf, giH s (S1 ) =
ξ∈Z
For n ∈ N, let us define the orthogonal projection Pn : H s (S1 ) → H s (S1 ) by
n
X
Pn f :=
fb(ξ) eξ ,
ξ=−n+1
where eξ (x) = ei2πx·ξ . If s ≤ t then it is easy to show that
kf − Pn f kH s (S1 ) ≤ ns−t kf kH t (S1 ) .
In practice, however, we are able to sample the values of a function f only
at finitely many points of S1 , and so we cannot even calculate Pn f ; hence let
us define the interpolation projection Qn : C(S1 ) → C(S1 ) by
µ ¶
µ ¶
j
j
Qn = Pn Qn , (Qn f )
=f
, (0 ≤ j < 2n);
2n
2n
recall that we can identify S1 with [0, 1[. This indeed determines Qn : if
1/2 < t and f ∈ H t (S1 ) then
n
X
X
eξ
fb(ξ + 2nη),
Qn f =
η∈Z
ξ=−n+1
and if moreover 0 ≤ s ≤ t then
kf − Qn f kH s (S1 ) ≤ Ct ns−t kf kH t (S1 ) ,
where Ct < ∞ depends only on t, and lim
Ct = ∞; this result is optimal
1
t→ 2 +
d
(for a proof, see e.g. the lecture notes [8]). The Fourier coefficients Q
n f can
be computed efficiently by FFT, in time O(n log n).
4
3
Approximation on S2
For the unit sphere
S2 = {x = (x1 , x2 , x3 ) ∈ R3 : kxkR3 = (x21 + x22 + x23 )1/2 = 1}
we use the spherical coordinates x = x(θ, φ), where
x1 = cos(φ) sin(θ),
x2 = sin(φ) sin(θ),
x3 = cos(θ),
0 ≤ θ ≤ π, 0 ≤ φ < 2π. The inner product for L2 (S2 ) is
Z 2π Z π
f (x) g(x) sin(θ) dθ dφ.
(f, g) 7→ hf, giL2 (S2 ) :=
0
0
Applying the Gramm–Schmidt process to the sequence (z 7→ z l )∞
l=0 we ob2
tain the Legendre polynomials Pl forming an orthogonal basis for L ([−1, 1]):
µ ¶l
d
1
0
Pl (z) = Pl (z) := l
(z 2 − 1)l .
2 l! dz
For l ∈ N and m ∈ {0, . . . , l}, the associated Legendre function P lm is defined
by
µ ¶m
d
m
2 m/2
Pl0 (z),
Pl (z) := (1 − z )
dz
and furthermore
Pl−m (z) := (−1)m
(l − m)! m
P (z).
(l + m)! l
Then an orthonormal basis for L2 (S2 ) consists of the spherical harmonic
functions Ylm ∈ C ∞ (S2 ) (l ∈ N, |m| ≤ l),
s
(2l + 1)(l − m)! m
Ylm (x(θ, φ)) = (−1)m
Pl (cos(θ)) eimφ .
4π(l + m)!
For l ∈ N and m ∈ Z, |m| ≤ l, the Fourier coefficient fb(l, m) of a function
f ∈ C ∞ (S2 ) is
Z 2π Z π
m
b
f (l, m) := hf, Yl iL2 (S2 ) =
f (x) Ylm (x) sin(θ) dθ dφ.
0
0
The Fourier inverse transform is now
f=
∞ X
l
X
l=0 m=−l
fb(l, m) Ylm .
It is often convenient to define fb(l, m) := 0 whenever l < 0 or |m| > l.
The spherical harmonic Fourier series can be obtained also from the Fourier series on groups SO(3) or U(2); this process is canonical, owing to the
representation theory of the groups.
5
The Laplace operator ∆ on S2 is the angular part of the Laplace operator
∂x21 + ∂x22 + ∂x23 on R3 :
1
∂
∆f =
sin(θ) ∂θ
µ
∂
f
sin(θ)
∂θ
¶
+
∂2
1
f.
sin2 (θ) ∂φ2
Also ∆ can be obtained using the representation theory of the symmetry
group of S2 . Notice that Ylm is an eigenfunction of the Laplacian, such that
∆Ylm = −l(l + 1) Ylm . Let the test function space D(S2 ) be C ∞ (S2 ) endowed
with the standard Fréchet space structure. The Sobolev space H s (S2 ) is the
completion of the test function space in the norm
kf kH s (S2 ) :=
Ã
∞
X
l
¯
¯2
X
¯b
¯
¯f (l, m)¯
hli2s
l=0
m=−l
!1/2
,
where hli = max{1, l}. The inner product for H s (S2 ) is obvious. Let us
define the orthogonal projection PN : H s (S2 ) → H s (S2 ) by
Pn f :=
l
n−1 X
X
l=0 m=−l
fb(l, m) Ylm .
Lemma 1. If s ≤ t then
kf − Pn f kH s (S2 ) ≤ ns−t kf kH t (S2 ) .
Proof. This is a matter of a simple calculation:
kf − Pn f kH s (S2 )
=
s≤t
≤
Ã
∞
X
l=n
ns−t
≤
ns−t
=
s−t
n
l2(s−t) l2t
Ã
Ã
l
X
m=−l
∞
X
l=n
∞
X
l2t
l
X
m=−l
hli2t
l=0
kf kH t (S2 )
|fb(l, m)|2
|fb(l, m)|2
l
X
m=−l
!1/2
!1/2
|fb(l, m)|2
!1/2
2
Now we are facing the difficulty that a function f can be sampled only
at finitely many points. As in the case of numerical Fourier analysis on the
torus, we need interpolation projections Qn , and study their approximation
properties in the scale of Sobolev spaces. Moreover, it would be important
6
to have some efficient computation method for the approximated Fourier
d
coefficients Q
n f (l, m).
Actually, there is an FFT-type algorithm for computing the Fourier coefn−1 X
l
X
ficients of a function f =
fb(l, m) Ylm , where n ∈ N. Notice that here
l=0 m=−l
f has at most N := n2 non-zero Fourier coefficients; a straightforward calculation for all these coefficients would cost O(N 2 ) operations, but with the
FFT-type approach only O(N (log N )2 ) operations and memory is needed.
The inverse Fourier transform has similar properties. The key to an FFTtype algorithm on S2 is that
Ylm (x) = clm Plm (cos(θ)) eimφ ,
where the classical FFT on S1 can be applied to the exponential factor, and
that there is a recurrence formula
m
m
(l − m + 1) Pl+1
(z) − (2l + 1) z Plm (z) + (l + m) Pl−1
(z) = 0
allowing one to construct a “Fast Legendre Transform”. It can be said that
all this is possible due to the rich symmetries, and there has been research
on FFT-type algorithms on many compact groups [4].
Let b ∈ N and n = 2b . The sampling points on S2 for degree n are
(n)
(n)
(n)
xjk = x(θj , φk ),
(n)
θj
:= π
j
,
2n
(n)
φk := 2π
k
,
2n
(n)
where 0 ≤ j, k < 2n. Notice that x0k ∈ S2 is the north pole for every k, and
furthermore we will have sampling weight equal to 0 at the north pole; yet for
convenience we may think of 4n2 points, which of course holds asymptotically
as n → ∞. Proposition 1 collects some results from [1]:
Proposition 1. Let l0 ∈ N and m0 ∈ Z, |m0 | ≤ l0 . If f =
l
n−1 X
X
l=0 m=−l
then
fb(l0 , m0 ) =
fb(l, m) Ylm
2n−1 2n−1
³
´
1 X X ³ (n) ´ m0 ³ (n) ´
(n)
f
x
Y
sin
θ
x
×
j
jk
l0
jk
4n2 j=0 k=0
×
n−1
³
´
4X 1
(n)
sin (2l + 1)θj
.
π l=0 2l + 1
Notice that here we used (less than) 4n2 sampled values of f to compute
n2 Fourier coefficients fb(l, m). The sampling measure implicit here is
2n−1 2n−1
n−1
³
´ 4 X
³
´
1
1 X X
(n)
(n)
δx(n) sin θj
sin (2l + 1)θj
µn := 2
.
4n j=0 k=0 jk
π l=0 2l + 1
7
Let us define Qn : C(S2 ) → C(S2 ) by
Qn f := Pn (f µn ).
Trivially Pn Qn = Qn . By Proposition 1, we have Qn Pn = Pn . Hence Q2n =
Qn (Pn Qn ) = (Qn Pn )Qn = Pn Qn = Qn , i.e. Qn is a projection. We want to
estimate kf − Qn f kH s (S2 ) . First, let us present some auxiliary results.
Lemma 2.
µ
cn (l, m) =
(
1,
0,
when l = 0 = m,
when 0 < l < 2n or
m 6∈ 2nZ.
Here 2nZ = {2nk : k ∈ Z}. Moreover,
|c
µn (l, m)| ≤ C hli1/2 ,
where C < 2.043 is a constant.
Proof. The equations for the Fourier coefficients were proven in [1]. Let us
then establish the estimate. Let
(
1,
0 < θ < π,
sgn(θ) =
−1, −π < θ < 0.
Notice that in the L2 ([−π, π])-sense,
∞
4X 1
sgn(θ) =
sin((2l + 1)θ).
π l=0 2l + 1
Due to the Gibbs phenomenon, the partial sums of this Fourier series do not
converge in L∞ , but yet we may say that
¯
¯ n
¯
¯4 X 1
¯
¯
sin((2l
+
1)θ)
¯<c
¯
¯
¯π
2l + 1
l=0
for every n ∈ N, where c < 1.179 is a constant. Thereby
=
=
≤
≤
=
|c
µ (l , m )|
¯Zn 0 0 ¯
¯
¯
¯ Y m0 dµn ¯
¯
¯ 2 l0
¯
¯ S 2n−1 2n−1
n−1
¯ 1 X X
´ 4X
´
³
³
´¯
³
1
¯
(n)
(n)
(n) ¯
Yl0m0 xjk sin θj
sin (2l + 1)θj ¯
¯ 2
¯
¯ 4n
π l=0 2l + 1
j=0 k=0
¯ m
¯
c sup sup ¯Yl0 0 (x(θ, φ))¯
0≤θ≤π 0≤φ<2π
p
°
°
c 2l0 + 1 °Yl0m0 °L2 (S2 )
p
c 2l0 + 1.
In the last inequality we used Lemma 8 from [5]. Hence the desired estimate
is obtained
2
For a function or a distribution g on S2 , denote glm := gb(l, m) Ylm .
8
Lemma 3. Let f ∈ D(S2 ) and g ∈ D 0 (S2 ). Then f g ∈ D 0 (S2 ) satisfies
fcg(l, m) =
∞
X
l2 =0
l2 +l
X
X
m1 m2
f\
l1 gl2 (l, m).
m1 ,m2 :
|mj |≤lj
m1 +m2 =m
l1 =|l2 −l|
Proof. There is a product formula
Yl1m1 Yl2m2 =
lX
1 +l2
clm1 l12mL2 YLm1 +m2 ,
L=|l1 −l2 |
where
l1 l2 L
cm
=
1 m2
s
(2l1 + 1)(2l2 + 1) l1 ,l2 ,L l1 ,l2 ,L
C0,0,0 Cm1 ,m2 ,m1 +m2 ,
4π(2L + 1)
l1 l2 L
l1 ,l2 ,l3
=0
being a Wigner symbol, see [9]. We use the convention cm
Cm
1 m2
1 ,m2 ,m3
whenever L < |l1 − l2 | or L > l1 + l2 or |m1 + m2 | > L. Thus
so that
m1 m2
l1 l2 l
b
f\
b(l2 , m2 ),
l1 gl2 (l, m) = δm,m1 +m2 cm1 m2 f (l1 , m1 ) g
fcg(l, m) =
=
l1
l2
∞
∞
X
X
X
X
m1 m2
f\
l1 gl2 (l, m)
l1 =0 m1 =−l1 l2 =0 m2 =−l2
l1
∞ X
∞
X
X
l2
X
m1 m2
l1 l2 l
δm,m1 +m2 cm
f\
l1 gl2 (l, m)
1 m2
l1 =0 l2 =0 m1 =−l1 m2 =−l2
=
=
∞
X
l2 +l
X
X
l2 =0
l1 =|l2 −l|
m1 ,m2 :
|mj |≤lj
m1 +m2 =m
∞
X
l2 +l
X
X
l2 =0
l1 =|l2 −l|
l1 l2 l
cm
fb(l1 , m1 ) gb(l2 , m2 )
1 m2
m1 m2
f\
l1 gl2 (l, m)
m1 ,m2 :
|mj |≤lj
m1 +m2 =m
2
Lemma 4. Let f be a function, and let µn be the sampling measure presented above. Then
¯
¯
¯
¯
¯
¯ \
¯
m1 m2
−r
r
r+1/2 ¯ b
≤
C
hli
hl
i
hl
i
f
(l
,
m
)
f
µ
(l,
m)
¯ 1 1 ¯.
¯ l1 nl2
¯
r
1
2
for every r > 1, where Cr is a constant depending on r, lim Cr = ∞.
r→1+
9
2
Proof. The last claim follows simply because µm
nl2 ≡ 0 when m2 6∈ 2nZ.
Notice that H r (S2 ) ⊂ C(S2 ) if and only if r > dim(S2 )/2 = 1. For functions
g, h : S2 → C let us denote Mg h := gh. When r > 1, H r (S2 ) is a Banach
algebra when equipped with the norm
g 7→
sup
h∈H r (S2 ): khkH r (S2 )≤1
kghkH r (S2 ) = kMg kL(H r (S2 )) .
This Banach algebra norm is equivalent to the Sobolev norm:
kgkH r (S2 ) ≤ kMg kL(H r (S2 )) ≤ cr kgkH r (S2 ) .
Here cr is a constant, and lim cr = ∞. Thereby we get
r→1+
¯
¯2
¯ \
¯
m1 m2
1 m2 m 2
µnl2 )l kH r (S2 )
f
µ
(l,
m)
¯ l1 nl2
¯ = hli−2r k(flm
1
1 m2 2
≤ hli−2r kflm
µnl2 kH r (S2 )
1
2 2
= hli−2r kMflm1 µm
nl2 kH r (S2 )
1
2 2
≤ hli−2r kMflm1 k2L(H r (S2 )) kµm
nl2 kH r (S2 )
1
≤
c2r
hli
−2r
= c2r hli−2r
2 2
1 2
kflm
kH r (S2 ) kµm
nl2 kH r (S2 )
1
¯
¯2
¯
2r
2r ¯ b
µn (l2 , m2 )|2 .
hl1 i hl2 i ¯f (l1 , m1 )¯ |c
By Lemma 2 above, |c
µn (l2 , m2 )|2 ≤ C 2 hl2 i, and this concludes the proof 2
Theorem 1. If −1 < s < t and t > 7/2 then
kf − Qn f kH s (S2 ) ≤ cs,t,ε ns−t n4+ε kf kH t (S2 ) .
Here cs,t,ε < ∞ is a constant depending only on s, t and ε > 0.
Proof. First, H k+1+ε (S2 ) ⊂ C k (S2 ) ⊂ H k (S2 ) for every k ∈ N and ε > 0,
and furthermore H k+1 (S2 ) 6⊂ C k (S2 ). Therefore we assume that f ∈ H t (S2 )
for some t > 1. Trivially
kf − Qn f kH s (S2 ) = k(f − Pn f ) + (Pn f − Qn f )kH s (S2 )
≤ kf − Pn f kH s (S2 ) + kPn f − Qn f kH s (S2 ) .
Lemma 1 gave the optimal estimate for kf − Pn f kH s (S2 ) , and the difficulties
come when dealing with kPn f − Qn f kH s (S2 ) . Now
Qn f = Pn (f µn )
l
n−1 X
X
fd
µn (l, m) Ylm .
=
l=0 m=−l
10
When 0 ≤ l < n, we have
fd
µn (l, m) = fb(l, m) +
l2 +l
X
∞
X
l2 =2n l1 =l2 −l
X
m1 m2
f\
l1 µnl2 (l, m)
m1 ,m2 :
m2 ∈2nZ
|mj |≤lj
m1 +m2 =m
by Lemmata 2 and 3. Thereby
(Qn − Pn )f
∞
l
n−1 X
X
X
m
Yl
=
l=0 m=−l
l2 +l
X
l2 =2n l1 =l2 −l
X
m1 m2
f\
l1 µnl2 (l, m).
m1 ,m2 :
m2 ∈2nZ
|mj |≤lj
m1 +m2 =m
We have
k(Pn − Qn )f k2H s (S2 )
¯
¯
¯
¯ ∞
l2 +l
n−1
l
X
X
X ¯¯ X
2s
=
hli
¯
¯
l=0
m=−l ¯l2 =2n l1 =l2 −l
¯
¯

m1 ,m2 :
m2 ∈2nZ
|mj |≤lj
m1 +m2 =m
 ∞
l2 +l
l
X X
X

2s
hli
≤


l=0
m=−l l2 =2n l1 =l2 −l
n−1
X
X
X
¯2
¯
¯
¯
¯
¯
m1 m2
\
fl1 µnl2 (l, m)¯
¯
¯
¯
¯
m1 ,m2 :
m2 ∈2nZ
|mj |≤lj
m1 +m2 =m
2

¯
¯
¯ \
¯
1 m2
µnl2 (l, m)¯ ,
¯flm
1


and by Lemma 4 we get
k(Pn − Qn )f k2H s (S2 )
≤ Cr2

n−1
l
X
X
hli2s−2r
l=0
 ∞
l2 +l
X X



l2 =2n l1 =l2 −l
m=−l
X
hl1 ir hl2 ir+1/2
m1 ,m2 :
m2 ∈2nZ
|mj |≤lj
m1 +m2 =m
11
2

¯
¯
¯b
¯
¯f (l1 , m1 )¯ .


The Hölder inequality gives us the next estimate:
k(Pn − Qn )f k2H s (S2 )
Cr2
≤

n−1
X
hli
l
X
2s−2r
m=−l
l=0

l2 +l
∞
X
X



l2 =2n l1 =l2 −l
X
hl1 i2r−2t
m1 ,m2 :
m2 ∈2nZ
|mj |≤lj
m1 +m2 =m


l2 +l
∞
X
X

×

l2 =2n l1 =l2 −l
X
m1 ,m2 :
m2 ∈2nZ
|mj |≤lj
m1 +m2 =m



2r+1 
hl2 i
×




¯2 
¯
¯

¯
m1 m2
hl1 i2t ¯f\
l1 µnl2 (l, m)¯ 


for some t large enough. Moreover,
∞
X
l2 +l
X
X
m1 ,m2 :
m2 ∈2nZ
|mj |≤lj
m1 +m2 =m
l2 =2n l1 =l2 −l
≤
∞
X
l22r+1
l2 =2n
≤
∞
X
l2 +l
X
(1 + 2 min{l1 , l2 /(2n)}) l12r−2t
l1 =l2 −l
l22r+1
(1 + l2 /n)
k:=l2 −n
≤
≤
≤
∞
X
(2l + 1)
(2l + 1)
(2l + 1)
l2 +l
X
l12r−2t
l1 =l2 −l
l2 =2n
≤
hl1 i2r−2t hl2 i2r+1
l22r+1 (1 + l2 /n) (l2 − n)2r−2t
l2 =2n
∞
X
(k + n)2r+1 (2 + k/n) k 2r−2t
k=n
∞
X
22r+1 k 2r+1 (2 + k/n) k 2r−2t
k=n
4r−2t+2
ct,r hli n
,
where ct,r depends on t and r. In the previous calculation, we had to demand
that 4r − 2t + 2 < −1, i.e. t > 2r + 3/2. Since r > 1, we must have t > 7/2.
12
With this knowledge, we get
k(Pn − Qn )f k2H s (S2 )
≤ Cr2 ct,r n4r−2t+2
 ∞
l2 +l
X X



l2 =2n l1 =l2 −l
Cr2
ct,r n
m=−l
X
m1 ,m2 :
m2 ∈2nZ
|mj |≤lj
m1 +m2 =m
4r−2t+2
n−1
X
hli
2s−2r+1
m=−l
≤
Cr2
X
m1 ,m2 :
m2 ∈2nZ
|mj |≤lj
m1 +m2 =m
ct,r n
4r−2t+2
hli
2s−2r+1
l1
X
m1 =−l1
≤ 9
Cr2
ct,r n
4r−2t+2
∞
X
¯2
¯
¯
¯b
f
(l
,
m
)
¯ 1 1 ¯
n−1
X
hli
2s−2r+3
l=0
≤ 9 Cr2 ct,r n4r−2t+2
hl1 i2t
n−1
X
X
l2 :max{2n,l1 −l}≤l2 ≤l1 +l
hl1 i2t (2l + 1) ×
l1 =n+1
l=0
× (2l + 1)
∞
X
l1 =n+1
¯
¯2
¯b
¯
¯f (l1 , m1 )¯
n−1
X


¯
¯2 
¯b
¯
¯f (l1 , m1 )¯ 


hl1 i2t
l=0
l
X
l
X
hli2s−2r+1
l=0

=
n−1
X
∞
X
hl1 i
2t
l1 =n+1
hli2s−2r+3 kf k2H t (S2 ) .
l1
X
m1 =−l1
¯2
¯
¯
¯b
¯f (l1 , m1 )¯
l=0
We may consider r > 1 to depend
t. We must suppose that 2s−2r+3 >
P on 2s−2r+3
−1, i.e. s > r − 2 > −1. Then n−1
hli
≤ c0s−r n2s−2r+4 , so that
l=0
2
n2s−2t n2r+6 kf k2H t (S2 ) .
k(Pn − Qn )f k2H s (S2 ) ≤ Cs,t,r
Here 2r + 6 = 2(r + 3) > 2 · 4. This yields
0
k(Pn − Qn )f kH s (S2 ) ≤ Cs,t,ε
ns−t n4+ε kf kH t (S2 ) ,
where Ctε is a constant depending on t and ε > 0. Consequently, an estimate
for kf − Qn f kH s (S2 ) is established
2
13
4
Insufficiency of methods
It is also important to notice that the results in numerical Fourier analysis on
S2 above are quite robust; more delicate treatises will be needed. To demonstrate this, let us apply our technique on S2 to get estimates for interpolation
projections on S1 . The notation in this section is parallel to the S2 -case.
We identify S1 with [0, 1[⊂ R. Recall that here
n
X
Pn f =
ξ=−n+1
fb(ξ) eξ .
Lemma 1’. If s ≤ t then
kf − Pn f kH s (S1 ) ≤ ns−t kf kH t (S1 ) .
This result is optimal, but we will present non-optimal results, yet in
analogy to the interpolation on the sphere S2 — we must remember that the
main point of this section is to demonstrate how insufficient our methods for
S2 may be.
The sampling points are {j/(2n) : 0 ≤ j < 2n}, and each point carries
the equal weight 1/(2n). The corresponding sampling measure is
2n−1
1 X
δj/(2n) .
µn =
2n j=0
Let Qn f := Pn (f µn ). Now we can calculate all the Fourier coefficients of µn :
µ
cn (ξ) =
(
1, when ξ ∈ 2nZ,
0, when ξ 6∈ 2nZ.
But we want to demonstrate what could possibly cause bad estimates for S 2
in the previous section. Hence let us modestly state the following:
Lemma 2’.
Moreover,
µ
cn (ξ) =
(
1,
0,
when ξ = 0,
when ξ ∈
6 2nZ,
|c
µn (ξ)| ≤ 1.
For a distribution g ∈ D 0 (S1 ), let gξ = gb(ξ) eξ . We write the well-known
formula fcg(ξ) = (fb ∗ gb)(ξ) in the following form:
14
Lemma 3’. Let f ∈ D(S1 ) and g ∈ D 0 (S1 ). Then f g ∈ D 0 (S1 ) satisfies
fcg(ξ) =
X
X
f\
ξ1 gξ2 (ξ).
ξ2 ∈Z ξ1 =ξ−ξ2
The Fourier coefficients of Qn f can be written down precisely, but we
settle for a weaker result:
Lemma 4’. Let f be a function, and let µn be the sampling measure presented above. Then
where r > 1/2.
¯
¯
¯b ¯
f
(ξ
)
¯ 1 ¯,
¯
¯
¯d ¯
f
µ
(ξ)
¯ n ¯ ≤ Cr hξi−r hξ1 ir hξ2 ir
Proof. The operator norm endows H r (S1 ) with a Banach algebra structure
if and only if r > dim(S1 )/2 = 1/2. Now the estimation process goes as in
the proof of Lemma 4, and we notice that |c
µn (ξ2 )| ≤ 1 here
2
Theorem 1’. If −1/2 < s < t and t > 3/2 then
kf − Qn f kH s (S1 ) ≤ cs,t,ε ns−t n kf kH t (S1 ) .
Here ct,s,ε < ∞ is a constant depending only on s, t and ε > 0.
Proof. First,
kf − Qn f kH s (S1 ) ≤ kf − Pn f kH s (S1 ) + kPn f − Qn f kH s (S1 ) .
Lemma 1’ gave the optimal estimate for kf − Pn f kH s (S1 ) . We must estimate
kPn f − Qn f kH s (S1 ) , but not too well: we follow the path taken in the proof
of Theorem 1. If −n < ξ ≤ n then Lemma 2’ yields
fd
µn (ξ) = fb(ξ) +
X
X
f\
ξ1 µnξ2 (ξ).
ξ2 : |ξ2 |≥2n ξ1 =ξ−ξ2
Hence
(Qn − Pn )f =
n
X
ξ=−n+1
eξ
X
X
ξ2 : |ξ2 |≥2n ξ1 =ξ−ξ2
15
f\
ξ1 µnξ2 (ξ),
so that Lemma 4 gives
k(Pn − Qn )f k2H s (S1 )
¯
¯ X
n
X
¯
2s ¯
hξi ¯
=
¯ξ2 : |ξ2 |≥2n
ξ=−n+1

n
X
X
hξi2s 
≤
2
¯
X ¯¯
¯
¯f\
ξ1 µnξ2 (ξ)¯
ξ2 : |ξ2 |≥2n ξ1 =ξ−ξ2
ξ=−n+1
≤ Cr2
¯2
¯
X
¯
¯
f\
ξ1 µnξ2 (ξ)¯
¯
ξ1 =ξ−ξ2
n
X
ξ=−n+1

hξi2s−2r 
X
X
ξ2 : |ξ2 |≥2n ξ1 =ξ−ξ2
The Hölder inequality yields
k(Pn − Qn )f k2H s (S1 )

n
X
X
hξi2s 
≤
ξ=−n+1

Here
×
X
X
2
¯
¯
¯
¯
hξ1 ir hξ2 ir ¯f\
.
ξ1 µnξ2 (ξ)¯
ξ2 : |ξ2 |≥2n ξ1 =ξ−ξ2
X
X
hξ1 i2t
ξ2 : |ξ2 |≥2n ξ1 =ξ−ξ2
X

hξ1 i2r−2t hξ2 i2r  ×

¯2
¯
¯b ¯ 
¯f (ξ1 )¯ .
hξ1 i2r−2t hξ2 i2r ≤ Ct,r n4r−2t+1 ;
ξ2 : |ξ2 |≥2n ξ1 =ξ−ξ2
here 4r − 2t < −1, i.e. t > 2r + 1/2 > 3/2. Consequently,
k(Pn − Qn )f k2H s (S1 )
n
X
4r−2t+1
≤ ct,r n
hξi2s−2r
ξ=−n+1
X
ξ2 : |ξ2 |≥2n
≤ c0t,r n2(s−t) n2(r+1) kf k2H t (S1 ) ,
hξ1 i
2t
¯
¯
¯ b ¯2
¯f (ξ1 )¯
where 2s − 2r > −1, i.e. s > r − 1 > −1/2. Since r > 1/2, we have
r + 1 > 3/2. Thus we eventually get
kf − Qn f kH s (S1 ) ≤ ct,s,ε ns−t n3/2+ε kf kH t (S1 ) ,
where −1/2 ≤ s < t, t > 3/2 and ε > 0.
16
2
5
Conclusion
Theorem 1’ shows that our method does not give the optimal result even in
the extremely simple case of the circle S1 : We lost 3/2 + ε Sobolev space
degrees, and we had to suppose that f ∈ H t (S1 ) with t > 3/2, even though
actually t > 1/2 can be assumed. This may suggest that our results for the
2-sphere S2 are not the best possible.
Acknowledgements. The work was funded by the Academy of Finland.
References
[1] J. Driscoll, D. Healy: Computing Fourier transforms and convolutions on
the 2-sphere. Adv. Appl. Math. 15 (1994), 202–250.
[2] D. M. Healy, D. N. Rockmore, P. J. Kostelec, S. Moore: FFTs for the 2sphere — improvements and variations. J. Fourier Anal. Appl. 9 (2003),
341–385.
[3] S. Kunis, D. Potts: Fast spherical Fourier algorithms. J. Comp. Appl.
Math. 161 (2003), 75–98.
[4] D. K. Maslen: Efficient computation of Fourier transforms on compact
groups. J. Fourier Anal. Appl. 4 (1998), 19–52.
[5] C. Müller: Spherical Harmonics. Lecture Notes in Mathematics 17.
Springer-Verlag, Berlin - New York, 1966.
[6] V. Turunen: Pseudodifferential calculus on compact Lie groups and homogeneous spaces. Ph.D. Thesis, Helsinki Univ. Techn., 2001.
http://lib.hut.fi/Diss/2001/isbn9512256940/
[7] V. Turunen: Pseudodifferential calculus on the 2-sphere. Helsinki University of Technology, Research Report A462, 2003. To appear in Proceedings
of the Estonian Academy of Sciences. Physics. Mathematics.
http://www.math.hut.fi/reports/a462
[8] G. Vainikko: Periodic integral and pseudodifferential equations. Helsinki
University of Technology, Research Report C13, 1996.
[9] N. J. Vilenkin: Special Functions and the Theory of Group Representations. Amer. Math. Soc., Providence, Rhode Island, 1968.
17
(continued from the back cover)
A458
Tuomas Hytönen
Translation-invariant Operators on Spaces of Vector-valued Functions
April 2003
A457
Timo Salin
On a Refined Asymptotic Analysis for the Quenching Problem
March 2003
A456
Ville Havu , Harri Hakula , Tomi Tuominen
A benchmark study of elliptic and hyperbolic shells of revolution
January 2003
A455
Yaroslav V. Kurylev , Matti Lassas , Erkki Somersalo
Maxwell’s Equations with Scalar Impedance: Direct and Inverse Problems
January 2003
A454
Timo Eirola , Marko Huhtanen , Jan von Pfaler
Solution methods for R-linear problems in C n
October 2002
A453
Marko Huhtanen
Aspects of nonnormality for iterative methods
September 2002
A452
Kalle Mikkola
Infinite-Dimensional Linear Systems, Optimal Control and Algebraic Riccati
Equations
October 2002
A451
Marko Huhtanen
Combining normality with the FFT techniques
September 2002
A450
Nikolai Yu. Bakaev
Resolvent estimates of elliptic differential and finite element operators in pairs
of function spaces
August 2002
HELSINKI UNIVERSITY OF TECHNOLOGY INSTITUTE OF MATHEMATICS
RESEARCH REPORTS
The list of reports is continued inside. Electronical versions of the reports are
available at http://www.math.hut.fi/reports/ .
A463
Marko Huhtanen , Jan von Pfaler
The real linear eigenvalue problem in setcn
November 2003
A462
Ville Turunen
Pseudodifferential calculus on the 2-sphere
October 2003
A461
Tuomas Hytönen
Vector-valued wavelets and the Hardy space H 1 (Rn ; X)
April 2003
A460
Timo Eirola , Jan von Pfaler
Numerical Taylor expansions for invariant manifolds
April 2003
A459
Timo Salin
The quenching problem for the N-dimensional ball
April 2003
ISBN 951-22-6828-0
ISSN 0784-3143