Multiresolution on n-dimensional spheres

Abstract

In the present paper, multiscale systems of polynomial wavelets on an n-dimensional sphere are constructed. Scaling functions and wavelets are investigated,and their reproducing and localization properties and positive definiteness are examined. Decomposition and reconstruction algorithms for the wavelet transform are presented. Formulae for variances in space and momentum domain, as well as for the uncertainty product, of zonal functions overn-dimensional spheres are derived and applied to the scaling functions.

Full text

Kyushu J. Math. 70 (2016), 353–374
doi:10.2206/kyushujm.70.353
MULTIRESOLUTION ON
n-DIMENSIONAL SPHERES
I. IGLEWSKA-NOWAK
(Received 12 November 2015 and revised 16 February 2016)
Abstract. In the present paper, multiscale systems of polynomial wavelets on an n-
dimensional sphere are constructed. Scaling functions and wavelets are investigated,
and their reproducing and localization properties and positive definiteness are examined.
Decomposition and reconstruction algorithms for the wavelet transform are presented.
Formulae for variances in space and momentum domain, as well as for the uncertainty
product, of zonal functions over n-dimensional spheres are derived and applied to the scaling
functions.
1. Introduction
In the past two decades, several wavelet constructions over n-dimensional spheres were
developed, cf., e.g., [1, 8, 9], all of them being a generalization of the two-dimensional case.
Our aim in this paper is to propose another approach, based on the book chapter of Conrad and
Prestin [6]. Wavelets here are polynomials in spherical harmonics, a feature that, on the one
hand, results in an oscillatory behavior of the wavelets themselves, and, on the other hand,
allows us to define a multiresolution analysis (MRA) of sampling spaces over equiangular
grids. Other MRA constructions for the 2-sphere can be found, e.g., in [4, 13, 14, 22].
To the best of our knowledge, the present research is the first attempt to define an MRA
over Sn. Although the generalization of [6] is straightforward, it demanded much carefulness,
especially in the generalization of the Clenshaw–Curtis quadrature in Lemma 3.1 and in the
computation of constants in Theorem 3.3 and Theorem 3.4 (and, consequently, constants for
reconstruction and decomposition algorithms of Subsection 3.3), since the addition theorem
has been reprinted with a false constant since [3] and it has been corrected only recently [17,
formula (2)]. (Note that the mistake was caused by a change of normalization convention for
hyperspherical harmonics, from
Sn|Yk
l(x)|2dσ (x) =1
to 1
nSn|Yk
l(x)|2dσ (x) =1.
2010 Mathematics Subject Classification: Primary 42C40, 42C05.
Keywords: n-dimensional spheres; polynomial wavelets; multiresolution analysis; uncertainty
product.
c
2016 Faculty of Mathematics, Kyushu University

354 I. Iglewska-Nowak
For the two-dimensional sphere, one still uses the first one such that the formulae presented
in the numerous papers about analysis over S2remain unaltered.) We also derive a formula
for the computation of the uncertainty product of zonal Sn-functions, a generalization of
Fern´andez’ ideas from [11].
The paper is organized as follows. After a short background presentation in Section 2,
we introduce polynomial weighted scaling functions and wavelets, present algorithms for
reconstruction and decomposition of spherical signals and show the MRA property in
Section 3. Section 4 is devoted to the computation of the uncertainty product of the weighted
scaling functions and analysis of its asymptotic behavior. Finally, in Section 5 we discuss the
property of positive definiteness of the weighted scaling functions. Two technical lemmas are
postponed to the Appendix.
2. Preliminaries
By Snwe denote the n-dimensional unit sphere in (n +1)-dimensional Euclidean space Rn+1
with the rotation-invariant measure dσ normalized such that
n=Sn
dσ =2πλ+1
(λ +1),
where λand nare related by
λ=n−1
2.
The surface element dσ is explicitly given by
dσ =sinn−1ϑ1sinn−2ϑ2···sin ϑn−1dϑ1dϑ2···dϑn−1dϕ, (1)
where (ϑ1,ϑ
2,...,ϑ
n−1,ϕ)∈[0,π]n−1×[0,2π) are spherical coordinates satisfying
x1=cos ϑ1,
x2=sin ϑ1cos ϑ2,
x3=sin ϑ1sin ϑ2cos ϑ3,
.
.
.
xn−1=sin ϑ1sin ϑ2···sin ϑn−2cos ϑn−1,
xn=sin ϑ1sin ϑ2···sin ϑn−2sin ϑn−1cos ϕ,
xn+1=sin ϑ1sin ϑ2···sin ϑn−2sin ϑn−1sin ϕ.
x, yor x·ystands for the scalar product of vectors with origin in Oand an endpoint on
the sphere. As long as it does not lead to misunderstandings, we identify these vectors with
points on the sphere.
The scalar product of f, g ∈L2(Sn)is defined by
f, gL2(Sn)=1
nSn
f (x) g(x) dσ(x),
and by ◦we denote the induced L2-norm.

Multiresolution on n-dimensional spheres 355
Gegenbauer polynomials Cλ
lof order λ∈R, and degree l∈N0, are defined in terms of
their generating function
∞

l=0
Cλ
l(t)rl=1
(1−2tr +r2)λ,t∈[−1,1].
They are real-valued and for some fixed λ= 0 orthogonal to each other with respect to the
weight function (1−◦
2)λ−1
2,see[16, formula 8.939.8].
Let Qldenote a polynomial on Rn+1homogeneous of degree l, i.e., such that Ql(az) =
alQl(z) for all a∈Rand z∈Rn+1, and harmonic in Rn+1, i.e., satisfying
Ql(z) =
n+1

ι=1
∂2
∂z2
ι
Ql(z) =0;
then Yl(x) =Ql(x ),x∈Sn, is called a hyperspherical harmonic of degree l.Thesetof
hyperspherical harmonics of degree lrestricted to Snis denoted by Hl=Hl(Sn),andweset
m=m
l=0Hl.Hl-functions are eigenfunctions of the Laplace–Beltrami operator ∗:=
|Snwith eigenvalue −l(l +2λ) =−l(l +n−1), and further, hyperspherical harmonics
of distinct degrees are orthogonal to each other. The number of linearly independent
hyperspherical harmonics of degree lis equal to
N=N(n, l) =(2l+n−1)(l +n−2)!
(n −1)!l!,
and consequently
dim m=(n +2m)(n +m−1)!
n!m!=n+2m
n+mn+m
m.
We were not able to find the last result in the literature, therefore, a proof is given in the
Appendix (Lemma A.1).
The addition theorem states that
Cλ
l(x ·y) =λ
l+λ
N

κ=1
Yκ
l(x)Y κ
l(y), (2)
for any orthonormal set {Yκ
l}κ=1,2,...,N(n,l) of hyperspherical harmonics of degree lon Sn.In
this paper, we will be working with the orthogonal basis for L2(Sn)=∞
l=0Hl, consisting
of hyperspherical harmonics given by
Yk
l(x) =Ak
l
n−1

ν=1
C((n−ν)/2)+kν
kν−1−kν(cos ϑν)sinkνϑν·e±ikn−1ϕ(3)
with l=k0≥k1≥···≥kn−1≥0, kbeing a sequence (k1,...,±kn−1)of integer numbers,
and normalization constants Ak
l,see[2, 23].
Every L1(Sn)-function fcan be expanded into a Laplace series of hyperspherical
harmonics by
f∼∞

l=0
Yl(f ;x),
where Yl(f ;x) is given by
Yl(f ;x) =(λ)(l +λ)
2πλ+1Sn
Cλ
l(x ·y)f(y) dσ(y) =l+λ
λCλ
l(x ·◦), f .(4)

356 I. Iglewska-Nowak
For zonal functions (i.e., those depending only on ϑ1ˆe, x,whereˆeis the north-pole of the
sphere ˆe=(1,0,...,0)) we obtain the Gegenbauer expansion
f(cos ϑ1)=∞

l=0
f(l)Cλ
l(cos ϑ1)
with Gegenbauer coefficients

f(l) =c(l, λ) 1
−1
f(t)Cλ
l(t)(1−t2)λ−1/2dt, (5)
where cis a constant that depends on land λ.
For f, g ∈L1(Sn),wheregis zonal, their convolution f∗gis defined by
(f ∗g)(x) =1
nSn
f(y)g(x ·y) dσ (y ).
With this notation we have
Yl(f ;x) =l+λ
λ(f ∗Cλ
l)(x), (6)
i.e., the function ((l +λ)/λ)Cλ
lis the reproducing kernel for Hl.
Further, any function f∈L2(Sn)has a unique representation as a mean-convergent
series
f(x)=
l
k
ak
lYk
l(x), x ∈Sn,
where
ak
l=ak
l(f ) =1
nSn
Yk
l(x) f (x) dσ (x ) =Yk
l,f,(7)
for a proof see [23]. In analogy to the two-dimensional case, we call ak
lthe Fourier
coefficients of the function f.
We identify zonal functions with functions over the interval [−1,1], i.e., whenever it
does not lead to mistakes, we write
f(x)=f(cos ϑ1).
3. Polynomial wavelets and polynomial multiscale decomposition
3.1. The polynomial multiscale decomposition
In [6], polynomial approximation and polynomial multiscale decomposition of L2(S2)are
discussed. Similarly as in [6, Theorem 7], we can argue that for sampling spaces Vj,j∈N,
given by Vj:= mj−1,where(mj)j∈Nis a strictly monotonously increasing sequence of
positive integers, the family {Vj}∞
j=1is a multiscale decomposition of L2(Sn). For our further
investigations we set
mj=2j−1,
which yields
Vj=2j−1−1=
2j−1−1

l=0
Hl

Multiresolution on n-dimensional spheres 357
for j∈N. The dimension of Vjis given by
dim Vj=(n +2j−2)(n +2j−1−2)!
n!(2j−1−1)!.
Further, let
Ij:= {(l, k1,...,k
n−1):l=0,1,...,2j−1−1,l≥k1≥···≥kn−1≥0}
be the index set of hyperspherical harmonics which span Vj,and
Nj={(s1,s
2,...,s
n−1,t):sν=0,...,2jfor ν=1,...,n−1;
t=0,...,2j+1−1}.
We shall prove a sampling theoremanalogous to the one given in [18]. However, since the
surface element σcontains higher powers of sin ϑν, see (1), we need a generalized quadrature
formula.
LEMMA 3.1. Let fbe a polynomial of degree at most Mand α∈N.Then
π
0
f(cos ϑ) sinαϑdϑ=
M

u=0
χufcos uπ
M,(8)
where χu=u·ωuwith
0=M=1
2and u=1for u=1,...,M −1,
and
ωu=πα!
2α−1M
[M/2]

μ=0
2μ(−1)μcos(2μuπ/M )
(α/2−μ+1)(α/2+μ+1).
If α/2−μ+1∈−N, the summand is understood to be equal to 0.
Proof. Since fis a polynomial, f(cos ◦)has a finite cosine-series representation
f(cos ϑ) =
M

μ=0
aμcos(μϑ),
where coefficients aμcan be computed exactly by the type-I discrete cosine transform
with M+1 nodes
aμ=2μ
M
M

u=0
ufcos uπ
Mcos μuπ
M.
The integral of a single frequency cos(μϑ ) is given by
π
0
cos(μϑ) sinαϑdϑ=πα!cos(μπ /2)
2α((α −μ+2)/2)((α +μ+2)/2).
This expression is to be understood as 0 for μ>αwith the same parity, and it is equal to 0
for odd μ. Together we obtain
π
0
f(cos ϑ) sinαϑdϑ=πα!
2α−1M
M

u=0
ufcos uπ
M
·[M/2]

μ=0
2μ(−1)μcos(2μuπ/M)
(α/2−μ+1)(α/2+μ+1).2

358 I. Iglewska-Nowak
Remark. For α=1 this yields the Clenshaw–Curtis quadrature.
THEOREM 3.2. Let f∈Vjbe given. Then we have, for (l, k) ∈Ij,
ak
l=π
2j
(s,t )∈Njn−1

ν=1
χ(j)
sνf(xj
s,t) Y k
l(xj
s,t), (9)
with
xj
s,t =s1π
2j,..., sn−1π
2j,tπ
2j(10)
and weight functions χ(j)
sνas given in Lemma 3.1 with M=2jand ωsνcomputed for
α=n−ν.
The proof is analogous to the proof of the sampling theorem for the two-dimensional
case presented in [18].
Proof. By definition of Vj, it suffices to consider the functions f=Yk
l,(l, k) ∈Ij.Their
Fourier coefficients are given by
ak
l=Ak
lAk
l
n
n−1

ν=1π
0
C(n−ν)/2+k
ν
k
ν−1−k
ν
(cos ϑν)C
(n−ν)/2+kν
kν−1−kν(cos ϑν)sink
ν+kν+n−νϑνdϑν
·2π
0
ei(∓k
n−1±kn−1)ϕ dϕ, (11)
see (1), (3), and (7). Since |∓k
n−1±kn−1|is at most equal to 2j−2, we have
δ±k
n−1,±kn−1=1
2π2π
0
ei(∓k
n−1±kn−1)ϕ dϕ
=1
2j+1
2j+1−1

t=0
ei(∓k
n−1±kn−1)tπ /2j.(12)
Hence, for k
n−1= kn−1the theorem is proven.
Now, let ν∈{1,2,...,n−1}be fixed and suppose k
ν=kν. The product
C(n−ν)/2+kν
k
ν−1−kν(cos ϑν)C
(n−ν)/2+kν
kν−1−kν(cos ϑν)sin2kνϑν(13)
is a polynomial in cos ϑνof degree at most k
ν−1+kν−1, which is less than or equal to 2j−2.
Therefore, quadrature formula (8) with M=2jand α=n−νcan be applied and the integral
of (13) with weight sinn−νϑνis equal to
2j

sν=0
χsνC(n−ν)/2+kν
k
ν−1−kνcos sνπ
2jC(n−ν)/2+kν
kν−1−kνcos sνπ
2jsin2kνsνπ
2j.(14)
Since for λ= 0 the functions → Cλ
l(cos ◦)and → Cλ
l(cos ◦)are orthogonal to each other
with respect to the weight sin2λ◦, the sum (14) vanishes for k
ν−1= kν−1. Thus, by induction,
ak
lvanishes for (l,k
)= (l, k), and if the indices match, the integrals in (11) can be replaced
by the sums (12) and (14). This yields (9). 2

Multiresolution on n-dimensional spheres 359
Now we introduce weighted scaling functions from Vj,
ϕj(◦·xj
s,t):= 1
√2nj 
(l,k)∈Ij
Yk
l(xj
s,t)Yk
l.(15)
The addition theorem (2) yields
ϕj=1
√2nj
2j−1−1

l=0
l+λ
λCλ
l.(16)
The next theorem summarizes some of the properties of ϕj.
THEOREM 3.3. Let (s, t ) ∈Njand xj
s,t be given by (10). Then the following hold.
(1) The functions ϕj(◦·xj
s,t)are real-valued.
(2) The functions ϕj(◦·xj
s,t)have the reproducing property
ϕj(◦·xj
s,t), f = 1
√2nj f(xj
s,t)for all f∈Vj.(17)
(3) We have
ϕj(◦·xj
s,t)2=1
√2nj ϕj(1)=(n +2j−2)(n +2j−1−2)!
2nj n!(2j−1−1)!.
(4) The function ϕj(◦·xj
s,t)is localized around xj
s,t, i.e.,
ϕj(◦·xj
s,t)
ϕj(xj
s,t ·xj
s,t)=min{f: f∈Vj,f(x
j
s,t)=1}.
(5) The set span{ϕj(◦·xj
s,t):(s, t ) ∈Nj}is equal to Vj, and
f=√2j(n−2)π
(s,t )∈Njn−1

ι=1
χ(j)
sιf(xj
s,t)ϕ
j(◦·xj
s,t)(18)
for f∈Vj.
(6) The set n−1

ι=1
χ(j)
sι1/2
ϕj(◦·xj
s,t):(s, t ) ∈Nj
is a tight frame in Vj,
2nj −1π
(s,t )∈Njn−1

ι=1
χ(j)
sι|f, ϕj(◦·xj
s,t)|2=f2
for every f∈Vj.
(7) The relation
ak
l(ϕj(◦·xj
s,t)) =⎧
⎪
⎨
⎪
⎩
1
√2nj Yk
l(xj
s,t)for (l, k) ∈Ij,
0otherwise

360 I. Iglewska-Nowak
and the two-scale relation
ak
l(ϕj(◦·xj
s,t)) =√2nak
l(ϕj+1(◦·xj
s,t)) for (l, k) ∈Ij,
0otherwise
hold for the Fourier coefficients of ϕj.
(8) We have Sn
ϕj(x ·xj
s,t)dσ(x)=23−n(1+j/2)πn/2+1(2λ)
(λ) (λ +1/2)(λ +1).
Remark. It follows from property (3) that for large jthe functions ϕjhave approximately
equal L2-norm or, more exactly,
lim
j→∞ ϕj2=21−n
n!.
Proof. (1) This follows directly from the representation (16).
(2) Use the expression (16) for ϕjand the property (4) of the functions ((l +λ)/λ)Cλ
l.
(3) It follows from property (2) that
ϕj(◦·xj
s,t)2=1
√2nj ϕj(xj
s,t ·xj
s,t)=1
2nj
2j−1−1

l=0
l+λ
λCλ
l(1),
and further, by [16, formula 8.937.4], that
ϕj(◦·xj
s,t)2=1
2nj
2j−1−1

l=0
l+λ
λl+2λ−1
l
=1
2nj
2j−1−1

l=0
(2l+n−1)(l +n−2)!
(n −1)!l!.
Consequently, by Lemma A.1, we have
ϕj(◦·xj
s,t)2=(n +2j−2)(n +2j−1−2)!
2nj n!(2j−1−1)!.
(4) For all f∈Vjwith f(xj
s,t)=1wehave
1=
(l,k)∈Ij
ak
l(f ) Y k
l(xj
s,t).
The Cauchy–Schwarz inequality yields
1≤
(l,k)∈Ij|ak
l(f )|2·
(l,k)∈Ij|Yk
l(xj
s,t)|2,(19)
and the equality in (19) holds only for a function ˜
fsuch that (ak
l(˜
f))
(l,k)∈Ijand
(Yk
l(xj
s,t))(l,k)∈Ijare linearly dependent, i.e.,
ak
l(˜
f)=α Y k
l(xj
s,t)

Multiresolution on n-dimensional spheres 361
for all (l, k) ∈Ijand for some constant α∈C. Consequently, ˜
fis a (√2nj α/n)-multiple
of ϕj,andforallf∈Vjwith f(xj
s,t)=1wehave
f2≥˜
f2=



ϕj(◦·xj
s,t)
ϕj(1)


.
(5) Let f∈Vjbe given by its Fourier series,
f=
(l,k)∈Ij
ak
lYk
l.
The coefficients ak
lcan be computed by (9). Together with (15) this yields (18), i.e., Vjis
spanned by the set {ϕj(◦·xj
s,t):(s, t ) ∈Nj}.
(6) For the L2-norm of f∈Vjwe write
f2=f, f = 
(l,k)∈Ij
ak
lYk
l,f.
From (18) it follows that
f2=2j(n−2)/2π
(l,k)∈Ij
(s,t )∈Nj
ak
ln−1

ι=1
χ(j)
sιf(xj
s,t)Yk
l,ϕ
j(◦·xj
s,t).
Further, we apply (17) for f(x
s,t)and for Yk
l,ϕ
j(◦·xj
s,t), and we obtain
f2=2j(n−2)/2π
(l,k)∈Ij
(s,t )∈Nj
ak
ln−1

ι=1
χ(j)
sιϕj(◦·xj
s,t), f Yk
l(xj
s,t)
=2j(n−2)/2π
(s,t )∈Njn−1

ι=1
χ(j)
sιϕj(◦·xj
s,t), f 
(l,k)∈Ij
ak
lYk
l(xj
s,t).
The last sum in this expression is equal to f(xj
s,t), and using for it again the reproducing
property (17) we obtain
f2=2nj −1π
(s,t )∈Njn−1

ι=1
χ(j)
sι|ϕj(◦·xj
s,t), f |2.
(7) This follows directly from the representation (15).
(8) The integral over Snis invariant with respect to rotations and, therefore, we can fix
xj
s,t =ˆe. Thus,
Sn
ϕj(x ·xj
s,t)dσ(x)=n−1π
0
ϕj(cos ϑ1)sinn−1ϑ1dϑ1.
From (16) we conclude that
Sn
ϕj(x ·xj
s,t)dσ(x)=n−1
√2nj
2j−1−1

l=0
l+λ
λπ
0
Cλ
l(cos ϑ1)sinn−1ϑ1dϑ1,

362 I. Iglewska-Nowak
and further, since Cλ
0(t) =1(see[16, formula 8.937.3]),
Sn
ϕj(x ·xj
s,t)dσ(x)=n−1
√2nj
2j−1−1

l=0
l+λ
λ1
−1
Cλ
0(t) Cλ
l(t)(1−t2)λ−1
2dt
=2πλ+1
2
2(λ+1
2)j (λ +1
2)
2j−1−1

l=0
l+λ
λ·δl,0·21−2λπ(2λ)
(λ)(λ +1).
The last equality holds by [16, formula 8.939.6]. Consequently,
Sn
ϕj(x ·xj
s,t)dσ(x)=22−2λ−jλ−1
2jπλ+3
2(2λ)
(λ) (λ +1
2)(λ +1).2
3.2. The polynomial wavelet space
Wavelet spaces Wj,j∈N, are defined as the direct sum
Wj=
2j−1

l=2j−1
Hl.
Their dimensions are given by
dim Wj=1
n!(n +2j+1−2)(n +2j−2)!
(2j−1)!−(n +2j−2)(n +2j−1−2)!
(2j−1−1)!.
We introduce the weighted wavelets from Wj,
ψj(◦·xj+1
s,t ):= 1
√2nj 
(l,k)∈Ij+1\Ij
Yk
l(xj+1
s,t )Yk
l,(s,t)∈Nj+1.(20)
Consequently, by (2) we have
ψj=1
√2nj
2j−1

l=2j−1
l+λ
λCλ
l.(21)
Their properties are summarized in the next theorem.
THEOREM 3.4. Let (s, t ) ∈Nj+1and xj+1
s,t be given by (10). Then the following hold.
(1) The functions ψj(◦·xj+1
s,t )are real-valued.
(2) The functions ψj(◦·xj+1
s,t )have the reproducing property
ψj(◦·xj+1
s,t ), f = 1
√2nj f(xj+1
s,t )for all f∈Wj.(22)
(3) ϕjand ψjare orthogonal to each other,
ϕj(xj
s,t·◦), ψj(xj+1
s,t ·◦)=0for all (s,t
)∈Nj,(s,t)∈Nj+1.

Multiresolution on n-dimensional spheres 363
(4) We have
ψj(◦·xj+1
s,t )2
=1
√2nj ψj(1)
=1
2nj n!(n +2j+1−2)(n +2j−2)!
2n(2j−1)!−(n +2j−2)(n +2j−1−2)!
(2j−1−1)!.
(5) The function ψj(◦·xj+1
s,t )is localized around xj+1
s,t , i.e.,
ψj(◦·xj+1
s,t )
ψj(xj+1
s,t ·xj+1
s,t )=min{f: f∈Wj,f(x
j+1
s,t )=1}.
(6) The set span{ψj(◦·xj+1
s,t ):(s, t) ∈Nj+1}equals Wj, and
f=√2(n−2)j−2π
(s,t )∈Nj+1n−1

ι=1
χ(j+1)
sιf(xj+1
s,t )ψ
j(◦·xj+1
s,t )(23)
for f∈Wj.
(7) The set n−1

ι=1
χ(j)
sι1/2
ψj(◦·xj+1
s,t ):(s, t) ∈Nj+1
is a tight frame in Wj,
√2(n−2)j−2π
(s,t )∈Nj+1n−1

ι=1
χ(j+1)
sι|f, ψj(◦·xj+1
s,t )|2=f2
for every f∈Wj.
(8) The relation
ak
l(ψj(◦·xj+1
s,t )) =⎧
⎪
⎨
⎪
⎩
1
√2nj Yk
l(xj+1
s,t )for (l, k) ∈Ij+1\Ij,
0otherwise
and the two-scale relation
ak
l(ψj(◦·xj+1
s,t )) =ak
l(ϕj+1(◦·xj+1
s,t )) for (l, k) ∈Ij+1\Ij,
0otherwise
hold for the Fourier coefficients of ψj.
(9) We have Sn
ψj(x ·xj+1
s,t )dσ(x)=0.
Proof. Property (3) follows directly from the definitions of ϕjand ψj. Proof of the other
items is analogous to the proof of Theorem 3.3. 2
Remark. For large values of j,ψjvanishes in L2-norm; more exactly,
ψj2=O(2−j), j →∞.

364 I. Iglewska-Nowak
3.3. Algorithms for reconstruction and decomposition
Let vj+1∈Vj+1,vj∈Vj,andwj∈Wjbe given. According to (18) and (23), these functions
can be represented by
vj+1=√2(j+1)(n−2)π
(s,t )∈Nj+1n−1

ι=1
χ(j+1)
sιvj+1(xj+1
s,t )ϕ
j+1(◦·xj+1
s,t ),
vj=√2j(n−2)π
(s,t )∈Njn−1

ι=1
χ(j)
sιvj(xj
s,t)ϕ
j(◦·xj
s,t),
wj=√2j(n−2)−2π
(s,t )∈Nj+1n−1

ι=1
χ(j+1)
sιwj(xj+1
s,t )ψ
j(◦·xj+1
s,t ).
Decomposition of the function vj+1into functions vjand wj,
vj+1=vj+wj,
is unique and the Fourier coefficients ak
lsatisfy
ak
l(vj+1)=ak
l(vj)+ak
l(wj)for (l, k) ∈Ij+1,
0for(l, k) /∈Ij+1.
Consequently,
ak
l(vj)=ak
l(vj+1)for (l, k) ∈Ij,
ak
l(wj)=ak
l(vj+1)for (l, k) ∈Ij+1\Ij.
It follows from Theorem 3.2 that
vj(x) =
(l,k)∈Ij
ak
l(vj+1)Y k
l(x)
=π
2j
(l,k)∈Ij
(s,t )∈Nj
n−1

ι=1
χ(j)
sιvj+1(xj
s,t) Y k
l(xj
s,t)Yk
l(x) (24)
and
wj(x) =
(l,k)∈Ij+1\Ij
ak
l(vj+1)Y k
l(x)
=π
2j+1
(l,k)∈Ij+1\Ij
(s,t )∈Nj+1
n−1

ι=1
χ(j+1)
sιvj+1(xj+1
s,t ) Y k
l(xj+1
s,t )Yk
l(x) (25)
for all x∈Sn, and also particularly for x(j+1)
s,t with (s, t) ∈Nj+1. Now let R be the operator
R(v(j+1))=v(j )

Multiresolution on n-dimensional spheres 365
mapping
v(j+1)=(vj+1(xj+1
s,t ))(s,t)∈Nj+1
onto
v(j) =(vj(xj
s,t))(s,t )∈Nj
via (24), and analogously, let Q be the operator
Q(v(j+1))=w(j)
defined by (25) for
w(j) =(wj(xj+1
s,t ))(s,t)∈Nj+1.
The operators R and Q describe the decomposition of a function vj+1∈Vj+1.
Decomposition Algorithm
Input:
v(j+1)
Compute for i=j, j −1,...,1:
v(i) =R(v(i+1))
w(i) =Q(v(i+1))
Output:
v(1),w(i),i=1,2,...,j
The functions v1and wi,i=1,2,...,j, can be reconstructed from their samples v(1),
w(i),i=1,2,...,j, via (24) and (25).
The reconstruction of a function vj+1∈Vj+1,
vj+wj=vj+1,
is given by
v(j+1)=R∗(v(j) )+Q∗(w(j ))
with
(R∗(v(j) ))p,q =π
2j
(l,k)∈Ij
(s,t )∈Nj
n−1

ι=1
χ(j)
sι(v(j))s,t Yk
l(xj
s,t)Yk
l(xj+1
p,q )
and
(Q∗(w(j)))p,q =π
2j+1
(l,k)∈Ij+1\Ij
(s,t )∈Nj+1
n−1

ι=1
χ(j+1)
sι(w(j))s,t Yk
l(xj+1
s,t )Yk
l(xj+1
p,q ),
for (p, q ) ∈Nj+1. The operators R∗and Q∗are adjoints of R and Q, respectively.

366 I. Iglewska-Nowak
Reconstruction Algorithm
Input:
v(1),w(i) ,i=1,2,...,j
Compute for i=1,2,...,j:
v(i+1)=R∗(v(i) )+Q∗(w(i) )
Output:
v(j+1)
3.3.1. Multiresolution analysis. The sequence (Vj)∞
j=1is a multiscale decomposition
of L2(Sn), i.e.,
(1) Vj⊂Vj+1for j∈N,and
(2) closurej∈NVj=L2(Sn).
The first property results from the definition, and the second one holds true by the Stone–
Weierstrass theorem since Snis compact.
4. Uncertainty of weighted scaling functions
In order to give a deeper characterization of the ϕjand ψj, we compute their uncertainty
product. According to [20] the variances in the space and momentum domain of a C2(Sn)-
function fwith Snx|f(x)|2dσ(x) = 0aregivenby
varS(f ) =Sn|f(x)|2dσ (x)
Snx|f(x)|2dσ(x)2
−1
and
varM(f ) =−Sn∗f(x)·¯
f(x)dσ(x)
Sn|f(x)|2dσ(x) ,
where ∗is the Laplace–Beltrami operator on Sn; see also [15]and[11].
Definition 4.1. The quantity
U(f ) =varS(f ) ·varM(f )
is called the uncertainty product of f.
According to [20, Theorem 1.2], for SO(n)-invariant functions f∈L2(Sn)∩C2(Sn),
U(f ) ≥n
2
and the lower bound is optimal (see also [10] for the non-zonal case). It is the limiting value
for t→0 of the uncertainty product of the so-called Gaussian measures
Gλ
t=
Gλ
t

Gλ
t1
,
Gλ
t=∞

l=0
e−tl(l+2λ)/2Cλ
l
Cλ
l2
2
,
see [20, Proposition 3.3].

Multiresolution on n-dimensional spheres 367
LEMMA 4.2. Let a zonal L2(Sn)-function be given by its Gegenbauer expansion
f(t)=∞

l=0
f(l)C
λ
l(t).
Its variances in space and momentum domain are given by
varS(f ) =∞
l=0
λ
l+λl+2λ−1
l|
f(l)|2
∞
l=0l+2λ
lλ2[
f(l) 
f(l+1)+
f(l) 
f(l+1)]
(l+λ)(l+λ+1)2
−1,(26)
varM(f ) =∞
l=1
lλ(l+2λ)
l+λl+2λ−1
l|
f(l)|2
∞
l=0
λ
l+λl+2λ−1
l|
f(l)|2,(27)
whenever the series are convergent.
Proof. Since ((l +λ)/λ)Cλ
lis the reproducing kernel of Hl, we obtain
1
nSn|f(x)|2dσ(x) =f∗f(ˆe) =∞

l=0
f(l) Cλ
l∗∞

l=0
f(l)C
λ
l(1)
=∞

l=0
λ
l+λ|
f(l)|2Cλ
l(1).
For Cλ
l(1)we write l+2λ−1
l,see[16, formula 8.937.4], and obtain
1
nSn|f(x)|2dσ (x) =∞

l=0
λ
l+λl+2λ−1
l|
f(l)|2.
In order to compute Snx|f(x)|2dσ(x), denote by gthe function x→ xf ( x ) and by 
Cλ
l
the function t→ tC
λ
l(t).Wehave
1
nSn
x|f(x)|2dσ(x) =f∗g(ˆe) =∞

l=0
f(l) Cλ
l∗∞

l=0
f(l) 
Cλ
l(1).
Now, for calculation of 
Cλ
lwe use formulae 8.932.1 and 8.930 from [16]:
(l +1)C
λ
l+1(t) =2(l +λ)t Cλ
l(t) −(l +2λ−1)Cλ
l−1(t) for l≥1,
Cλ
1(t) =2λt =2λtC
λ
0(t),
and obtain
1
nSn
x|f(x)|2dσ (x) =1
2λ
∞

l=0
f(l)C
λ
l∗
f(0)C
λ
1(1)
+∞

l=0
f(l) Cλ
l∗∞

l=1
l+1
2(l +λ) 
f(l)C
λ
l+1(1)
+∞

l=0
f(l) Cλ
l∗∞

l=1
l+2λ−1
2(l +λ) 
f(l)C
λ
l−1(1).

368 I. Iglewska-Nowak
Further, by the reproducing kernel property of ((l +λ)/λ) Cλ
lwe get
1
nSn
x|f(x)|2dσ (x) =1
2(λ +1)
f(1)
f(0)C
λ
1(1)
+∞

l=1
λ(l +1)
2(l +λ)(l +λ+1)
f(l+1)
f(l) Cλ
l+1(1)
+∞

l=1
λ(l +2λ−1)
2(l +λ)(l +λ−1)
f(l−1)
f(l) Cλ
l−1(1),
and consequently, by index-shift l→ l−1 in the second series,
1
nSn
x|f(x)|2dσ (x) =λ
λ+1
f(1)
f(0)
+∞

l=1
λ(l +1)
2(l +λ)(l +λ+1)l+2λ
l+1
f(l +1)
f(l)
+∞

l=0
λ(l +2λ)
2(l +λ+1)(l +λ) l+2λ−1
l
f(l) 
f(l+1).
Altogether we have
1
nSn
x|f(x)|2dσ(x)
=∞

l=0
λ2
(l +λ)(l +λ+1)l+2λ
l[
f(l +1)
f(l)+
f(l) 
f(l +1)].
The same formula (up to complex conjugation and for zero-mean functions) is derived in [7]
with slightly different methods.
Finally, since
−∗f=∞

l=1
l(l +2λ) 
f(l) Cλ
l,
we have
−1
nSn
∗f(x)·¯
f(x)dσ(x)=∞

l=1
l(l +2λ) 
f(l)C
λ
l∗∞

l=0
f(l)C
λ
l(1)
=∞

l=1
l(l +2λ)|
f(l)|2·λ
l+λCλ
l(1)
=∞

l=1
lλ(l +2λ)
l+λl+2λ−1
l|
f(l)|2.
Combining the above formulae we obtain (26) and (27). 2
Remark. For λ=1
2these formulae reduce to those obtained in [11, Section 5.2].
Now, we want to compute the space and momentum variances for the weighted scaling
functions. However, in order to obtain more general results, we do not restrict ourselves to

Multiresolution on n-dimensional spheres 369
the scales 2j−1−1. Let us introduce the notation
m=C(m) ·
m

l=0
l+λ
λCλ
l,
where C(m) is a constant.
PROPOSITION 4.3. The variances in space and momentum domain of m,m∈N, are equal
to
varS(m)=2m+2λ+1
2m2
−1,(28)
varM(m)=m(m+2λ+1)(2λ+1)
2λ+3.(29)
Proof. In the case of weighted scaling functions we have 
f(l) =((l +λ)/√2nj λ) for l≤
m=2j−1−1. With these values of Gegenbauer coefficients we obtain
2nj
nSn|f(x)|2dσ(x) =2nj ·
m

l=0
λ
l+λl+2λ−1
l|
f(l)|2
=
m

l=0
l+λ
λl+2λ−1
l=(2m+2λ+1)(m+2λ)!
m!(2λ+1)!,
see Lemma A.1, and
2nj
nSn
x|f(x)|2dσ (x) =2nj ·
m−1

l=0l+2λ
lλ2[
f(l) 
f(l+1)+
f(l) 
f(l+1)]
(l +λ)(l +λ+1)
=2·
m−1

l=0l+2λ
2λ=2m+2λ
2λ+1,
see [16, formula 0.15.1], and finally
−2nj
nSn
∗f(x)·¯
f(x)dσ(x)=2nj ·
m

l=0
lλ(l +2λ)
l+λ|
f(l)|2l+2λ−1
l
=2(2λ+1)
m

l=1
(l +λ) l+2λ
l−1
=2(2m+2λ+1)(m+2λ+1)!(2λ+1)(λ+1)
(m −1)!(2λ+3)!,
see Lemma A.2. Combining these expressions together according to (26) and (27) yields (28)
and (29), respectively. 2
Values of space variance, momentum variance and uncertainty product for several values
of λand mare collected in Table 1. Note that the lower bound for U(f ) is equal to λ+1
2.

370 I. Iglewska-Nowak
TABLE 1. Space variance, momentum variance and uncertainty product of min various dimensions.
varS(m)
varM(m)
U(m)λ=1
2λ=1λ=3
2λ=2λ=5
2λ=3
m=1 3 5.25 8 11.3 15 19.3
1.5 2.4 3.33 4.29 5.25 6.22
2.12 3.55 5.16 6.94 8.87 10.94
m=2 1.25 2.06 3 4.06 5.25 6.56
4 6 8101214
2.24 3.52 4.9 6.37 7.94 9.59
m=3 0.781.251.782.363 3.69
7.5 10.8 14 17.14 20.25 23.33
2.42 3.67 4.99 6.36 7.79 9.29
m=4 0.560.891.251.642.062.52
12 16.8 21.33 25.71 30 34.22
2.6 3.87 5.16 6.5 7.87 9.28
m=5 0.440.690.961.251.561.89
17.5 24 30 35.71 41.25 46.67
2.78 4.07 5.37 6.68 8.02 9.39
m=6 0.360.560.781.011.251.51
24 32.4 40 47.14 54 60.67
2.94 4.27 5.58 6.89 8.22 9.56
m=7 0.310.470.650.841.041.25
31.5 42 51.33 60 68.25 76.22
3.11 4.46 5.79 7.11 8.43 9.76
m=15 0.14 0.21 0.28 0.36 0.44 0.52
127.5 162 190 214.29 236.25 256.67
4.19 5.83 7.35 8.8 10.2 11.57
m=31 0.07 0.1 0.13 0.17 0.2 0.24
511.5 632.4 723.33 797.14 860.25 916.22
5.79 7.92 9.82 11.57 13.21 14.78
m=63 0.03 0.05 0.06 0.08 0.1 0.11
2047.5 2494.8 2814 3060 3260.25 3430
8.01 10.99 13.47 15.74 17.83 19.79
m=127 0.02 0.02 0.03 0.04 0.05 0.06
8191.5 9906 11 091.3 11 974.3 12 668.3 13 236.2
11.38 15.34 18.76 21.82 24.61 27.2
m=255 0.01 0.01 0.02 0.02 0.02 0.03
32 767.5 39 474 44 030 47 357.1 49 916.3 51 963
16.05 21.58 26.33 30.55 34.37 37.9

Multiresolution on n-dimensional spheres 371
LEMMA 4.4. The space variance, momentum variance and uncertainty product of m
behave for m→∞like
varS(m)∼2λ+1
m,
varM(m)∼2λ+1
2λ+3·m2,
U(m)∼2λ+1
√2λ+3·m1/2.
Proof. By elementary calculations. 2
Remark. For λ=1
2this coincides with the result obtained by Fern´andez in [12, Section 5.2].
In a similar way, we can describe the behavior of the mfor a growing space dimension,
as described in the next lemma.
LEMMA 4.5. The space variance, momentum variance and uncertainty product of m
behave for λ→∞like
varS(m)∼⎧
⎨
⎩
4λ2for m=1,
1
m2·λ2for m>1,
varM(m)∼2mλ,
U(m)∼⎧
⎪
⎨
⎪
⎩
(2λ)3/2for m=1,
2
m1/2
·(λ)3/2for m>1.
Proof. By elementary calculations. 2
5. Spherical basis functions
Definition 5.1. A continuous function G:[−1,1]→Ris said to be positive definite on Sn
if, for every L∈Nand any sequence of points (xl)L
l=1⊆Sn, the corresponding Gramian
matrix A=(G(xj·xk))j=1,...,L
k=1,...,L
is positive semidefinite. Further, if Ais positive definite,
then Gis said to be strictly positive definite.
Positive definite functions and strictly positive definite functions are used for
interpolation scattered data on a sphere, and strict positive definiteness is much more
advantageous for stability of algorithms; see [5, 19, 24] for a discussion. The starting point
in determining whether a function is positive definite or not is the classical result obtained by
Schoenberg in [21].
THEOREM 5.2. A continuous function G:[−1,1]→Ris positive semidefinite on Snif and
only if it has the form
G=∞

l=0
alCλ
l
with al≥0and ∞
l=0Cλ
l(1)<∞.

372 I. Iglewska-Nowak
Further, for positive definiteness of cardinality L(i.e., for some fixed Linstead of L∈N,
see [19]) we have the following sufficient condition, proven in [24].
THEOREM 5.3. Let Gsatisfy the conditions of Theorem 5.2 and let (xl)L
l=1⊆Sn.In
order that the matrix A=(G(xj·xk))j=1,...,L
k=1,...,L
is positive definite it is sufficient that the
coefficients albe positive for 0≤l<L.
Finally, a characterization of strict positive definiteness is given in [5].
THEOREM 5.4. Let Gsatisfy the conditions of Theorem 5.2. In order that Gis strictly
positive definite on Snit is necessary and sufficient that infinitely many coefficients alwith
odd index land infinitely many coefficients alwith even index lbe positive.
According to these theorems, the weighted scaling functions φjare positive definite,
strictly positive definite of cardinality 2j−1, but they are not strictly positive definite.
A. Appendix
LEMMA A.1. Fo r n≥2,
m

l=0
(2l+n−1)(l +n−2)!
(n −1)!l!=(n +2m)(n +m−1)!
n!m!
holds.
Proof. (By induction) It is straightforward to verify that the formula holds for m=0. Suppose
that it is valid for some m.Form+1, we have
m+1

l=0
(2l+n−1)(l +n−2)!
(n −1)!l!=(n +2m)(n +m−1)!
n!m!+(n +2m+1)(n +m−1)!
(n −1)!(m +1)!
=[(m +1)(n +2m) +n(n +2m+1)]·(n +m−1)!
n!(m +1)!
=(n +2m+2)(n +m) ·(n +m−1)!
n!(m +1)!
=(n +2m+2)(n +m)!
n!(m +1)!.2
LEMMA A.2. Fo r λ>0, we have
m

l=1
(l +λ) l+2λ
l−1=(2m+2λ+1)(m+2λ+1)!(λ +1)
(m −1)!(2λ+3)!.

Multiresolution on n-dimensional spheres 373
Proof. (By induction) It is straightforward to verify that the formula holds for m=1.
Suppose, it is valid for some m.Form+1, we have
m+1

l=1
(l +λ) l+2λ
l−1
=(2m+2λ+1)(m+2λ+1)!(λ +1)
(m −1)!(2λ+3)!+(m +λ+1)(m+2λ+1)!
m!(2λ+1)!
=(m +2λ+1)!m(2m+2λ+1)(λ+1)+(m +λ+1)(2λ+2)(2λ+3)
m!(2λ+3)!
=(m +2λ+2)!(2m+2λ+3)(λ+1)
m!(2λ+3)!.2
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I. Iglewska-Nowak
West Pomeranian University of Technology in Szczecin
School of Mathematics
al. Piast´
ow 17
70–310 Szczecin
Poland
(E-mail: ilona.iglewska-nowak@zut.edu.pl)