A simple Proof of Stolarsky's Invariance Principle

0.16 MB·English

by  Johann S. Brauchart

A simple Proof of Stolarsky’s
Invariance Principle
∗
Johann S. Brauchart†and Josef Dick‡
1
School of Mathematics and Statistics,
1
0 University of New South Wales,
2 Sydney, NSW, 2052, Australia

n [email protected],
a
J [email protected]

4
2

]
A Abstract
N
Stolarsky [Proc. Amer. Math. Soc. 41 (1973), 575–582] showed a beautiful relation
h. that balances the sums of distances of points on the unit sphere and their spherical cap L2-
t discrepancy to give the distance integral of the uniform measure on the sphere a potential-
a
theoreticalquantity(Bjo¨rck[Ark. Mat. 3(1956),255–269]). Readdifferentlyitexpressesthe
m
worst-case numerical integration error for functions from the unit ball in a certain Hilbert
[ space setting in terms of the L2-discrepancy and vice versa (first author and Womersley

1 [Preprint]). In this note we givea simple proofofthe invariance principle using reproducing
v kernel Hilbert spaces.
8
4
4 1 Introduction
4
.
1
We consider the unit sphere
0
1
1 Sd = z = (z ,...,z ) Rd+1 : z = z2+ +z2 = 1
: 1 d+1 ∈ k k 1 ··· d+1
v
n q o
Xi embedded in the Euclidean space Rd+1, d 2. Let f : Sd C be a continuous function. Then
≥ →
r we approximate the integral f(x) dσ (x), where σ is the normalized Lebesgue surface area
a Sd d d
measure on Sd ( dσ = 1), by an equal weight numerical integration rule
Sd d R
R
N−1
1
Q (f)= f(z ) (1)
N k
N
k=0
X
where z ,...,z Sd are the integration nodes on the sphere. In order to analyze the
0 N−1
∈
integration error committed by the approximation, we define a worst-case error by
N−1
1
e( ,Q ) = sup f(x)dσ (x) f(z ) ,
N d k
H f∈H,kfk≤1(cid:12)(cid:12)ZSd − N Xk=0 (cid:12)(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
where denotes a normed function space(cid:12)with norm . The rate of de(cid:12)cay of the worst-case
H k·k
error depends on the function space and the integration nodes. For a fixed function space, the
∗MSC: primary 41A30; secundary 11K38, 41A55.
†The author is supported by an APART-Fellowship of theAustrian Academy of Sciences.
‡The author is supported by an Australian Research Council Queen Elizabeth 2 Fellowship.
1
worst-case error can serve as a quality criterion for different sets of integration nodes, meaning
that the performance of a set of N quadrature points z ,...,z can be compared to another
0 N−1
set of N quadrature points by comparing the corresponding worst-case errors. Generally, this
only means that the integration error is smaller, but sometimes the worst-case error allows also
a geometrical interpretation.
Another quality criterion for points on the sphere exploits the potential energy, or more
generally, theRieszs-energy ofconfigurations ofpointsmodelingunitcharges whicharethought
to interact through a potential 1/ s (s = 0), where denotes the Euclidean distance.(We
k·k 6 k·k
refer the reader to the survey papers [10] and [13] and for universally optimal configurations to
[8].) A particular instance is the (normalized) sum of distances (s = 1)
−
N−1
1
z z , z ,...,z Sd.
N2 k k − ℓk 1 N ∈
k,ℓ=0
X
It is well-known from potential theory (see Bjo¨rck [4]) that this (discrete) sum of distances of
optimal N-point configurations approaches the associated (continuous) distance integral of the
uniform measure σ on Sd as N . In fact, any sequence of N-point systems with this
d
→ ∞
property turns out to be ’asymptotically uniformly distributed’; that is, the discrete probability
measure obtained by placing equal charges at the points tends to the uniform measure in the
weak-star sense. The difference
N−1
1
z x dσ (z)dσ (x) z z (2)
Sd Sdk − k d d − N2 k k − ℓk
Z Z k,ℓ=0
X
measuring the deviation between theoretical and empirical ( 1)-energy quantifies the quality
−
of points on the sphere (and, indirectly, their uniform distribution) using energy. It should be
mentioned that the upper bound of correct order N−1−1/d for (2) (for optimal configurations)
was obtained by Stolarksy [16] using his invariance principle and a result of Schmidt [14] on
the discrepancy of spherical caps. The correct-order lower bound (N−1−1/d) was established by
Beck [3] using his Fourier transform technique.
The spherical cap discrepancy measures the maximum deviation between theoretical and
empirical distribution with respect to spherical caps as test sets. It can be used to compare
point sets on the sphere with respect to their distribution properties. To introduce the concept
of spherical cap discrepancy, we require some notation. A spherical cap centered at x Sd with
∈
’height’ t [ 1,1] is the set
∈ −
C(x;t)= z Sd : x,z t .
{ ∈ h i ≥ }
The family of all spherical caps is denoted by
C = C(x;t) :x Sd,t [ 1,1] .
{ ∈ ∈ − }
For a set J Rd+1 we define the indicator function
⊆
1 if x J,
1 (x)= ∈
J
(0 otherwise.
For a measurable set J Sd let
⊆
σ (J) = 1 (x) dσ (x).
d J d
Sd
Z
2
Using spherical caps we can define another quality criterion for points on the sphere Sd in terms
of their distribution properties. One such criterion is the spherical cap L -discrepancy, which is
2
given by
2 1/2
1 1 N−1
L (P) = σ (C(z;t)) 1 (z ) dσ (z)dt ,
2 d C(z;t) k d
Z−1ZSd(cid:12)(cid:12) − N Xk=0 (cid:12)(cid:12) 
(cid:12) (cid:12)
where P = z ,...,z  . (cid:12) (cid:12) 
0 N−1 (cid:12) (cid:12)
{ }
We considered three, seemingly different, measures which can be applied to point sets on
the sphere. It turns out that in some instances, the three measures are related to each other.
Stolarksy’s insight [16] was that the sum of distances of points on the sphere and the spherical
cap discrepancy coincide. On the other hand, also the sum of distances of points on the sphere
and the worst-case error coincide for a certain choice of function space, see [7] and also Sloan
and Womersley [15] regarding a generalized discrepancy of Cui and Freeden [9].
In this paper we give a simple proof of these results based on reproducing kernel Hilbert
spaces. We also provide some generalizations which follow from our approach.
2 Reproducing kernel Hilbert space
We define a reproducing kernel Hilbert space using the general approach of [12, Ch. 9.6].
For x,y Sd we define the function KC : Sd Sd R by
∈ × →
1
KC(x,y)= 1C(z;t)(x)1C(z;t)(y) dσd(z) dt. (3)
Z−1ZSd
Since 1 (x) = 1 (z), we also have
C(z;t) C(x;t)
1
KC(x,y) = 1C(x;t)(z)1C(y;t)(z) dσd(z) dt.
Z−1ZSd
The function is obviously symmetric, i.e. we have KC(x,y) = KC(y,x). Further, let
a ,...,a C and x ,...,x Sd. Then we have
0 N−1 0 N−1
∈ ∈
N−1 1 N−1
akaℓKC(xk,xℓ) = akaℓ1C(x;t)(xk)1C(x;t)(xℓ) dσd(x) dt
k,ℓ=0 Z−1ZSdk,ℓ=0
X X
2
1 N−1
= a 1 (x ) dσ (x) dt
k C(x;t) k d
Z−1ZSd(cid:12)(cid:12)Xk=0 (cid:12)(cid:12)
(cid:12) (cid:12)
0. (cid:12) (cid:12)
≥ (cid:12) (cid:12)
Thus, the function KC is symmetric and positive definite. By [2], this implies that KC is a
reproducing kernel. It is also shown in [2] that a reproducing kernel uniquely defines a Hilbert
space of functions with a certain inner product. Let C = (KC,Sd) denote the corresponding
H H
reproducing kernel Hilbert space of functions f :Sd R with reproducing kernel KC.
→
We consider now functions f ,f : Sd C which permit a certain integral representation.
1 2
→
Let g ,g : Sd [ 1,1] C with g ,g L (Sd [ 1,1]) and
1 2 1 2 2
× − → ∈ × −
1
f (x) = g (z;t)1 (x) dσ (z) dt, i= 1,2. (4)
i i C(z;t) d
Z−1ZSd
3
Notice that for any fixed y Sd the function KC(,y) also is of this form, where the function g
∈ ·
is given by 1 (y) (considered as a function of z and t and where y is fixed). For functions
C(z;t)
of this form we can define an inner product by
1
f ,f = g (z;t)g (z;t)dσ (z)dt. (5)
h 1 2iKC Z−1ZSd 1 2 d
Let y Sd be fixed. With this definition we obtain
∈
1
hf1,KC(·,y)iKC = Z−1ZSdg1(z;t)1C(z;t)(y) dσd(z) dt = f1(y).
By [2], the inner product in C is unique. Therefore, functions fi, which are given by (4) and
H
for which hfi,fiiKC < ∞, are in HC and (5) is an inner product for those functions in HC.
Consider now the reproducing kernel KC. We have
1 min{hx,zi,hy,zi}
1 (x)1 (y) dt = dt = 1+min x,z , y,z .
C(z;t) C(z;t)
{h i h i}
Z−1 Z−1
Thus
1
KC(x,y) = 1C(z;t)(x)1C(z;t)(y) dσd(z) dt
Z−1ZSd
= 1+ min x,z , y,z dσ (z).
d
Sd {h i h i}
Z
We have
1
min x,z , y,z = [ x,z + y,z x y,z ]
{h i h i} 2 h i h i−|h − i|
and
x,z dσ (z)= 0.
d
Sdh i
Z
If x = y, then we therefore obtain
min x,z , y,z dσ (z) = 0.
d
Sd {h i h i}
Z
Let now x = y. Then
6
1
min x,z , y,z dσ (z) = x y,z dσ (z)
d d
Sd {h i h i} −2 Sd|h − i|
Z Z
1 x y
= x y − ,z dσ (z).
−k − k 2ZSd(cid:12)(cid:28)kx−yk (cid:29)(cid:12) d
(cid:12) (cid:12)
The last integral does not depend on the unit vector (x y(cid:12))/ x y by r(cid:12)otational symmetry.
− (cid:12) k − k (cid:12)
Thus, we have (cf. Appendix A)
1 1ω (Bd) 1Γ((d+1)/2) 1
C := p,z dσ (z) = d−1 = Hd = as d . (6)
d 2 Sd|h i| d d ωd d(Sd) d √πΓ(d/2) ∼ √2πd → ∞
Z H
(Here, ω is the surface area of Sd, Bd is the unit ball in Rd, is the d-dimensional Hausdorff
d d
H
measure normalized such that the d-dimensional unit cube [0,1)d has measure one and Γ(z) is
the Gamma function.)
Therefore we obtain the following closed form representation:
KC(x,y) = 1 Cd x y . (7)
− k − k
The reproducing kernel has the following properties: for y,z Sd we have
∈
4
1 2C K(x,y) 1;
d
• − ≤ ≤
K(x,y) =1 2C if and only if x = y (clearly, 0 < 2C < 1);
d d
• − −
K(x,y) =1 if and only if x = y;
•
Note that the Karhunen-Loevy expansion of the function x y is based on ultraspherical
k − k
harmonics. HencetheeigenfunctionsofKC aretheultrasphericalharmonics. Thecorresponding
eigenvalues arealsoknown. Therefore,thefunctionsin C canbeexpandedusingultraspherical
H
harmonics and the inner product can be written using the coefficients of such an expansion. See
[7] for these results.
3 Worst-case error
Let kfkKC = hf,fiKC denote the norm in HC. Then we define the worst-case error for a
quadrature rule Q given in (1) by
p N
e(HC,QN) = sup(cid:26)(cid:12)ZSdf(x) dx−QN(f)(cid:12) :f ∈ HC,kfkKC ≤ 1(cid:27).
(cid:12) (cid:12)
Let f ∈ HC. Then, by the repro(cid:12)(cid:12)ducing kernel propert(cid:12)(cid:12)y f(y) = hf,KC(·,y)iKC for y ∈ Sd,
and, since the integration functional f 7→ Sdf dσd is bounded on HC and has the ’representer’
SdKC(·,z)dσd(z), one can write R
R
N−1
1
Sdf(x) dσd(x)− N f(xk)= hf,R(HC,QN;·)iKC ,
Z k=0
X
where the ’representer’ of the error of numerical integration for the rule QN for functions in C
H
is given by
N−1
1
( C,QN;x) = KC(x,y) dσd(y) KC(x,xk), x Sd.
R H Sd − N ∈
Z k=0
X
The Cauchy-Schwarz inequality yields
N−1
1
(cid:12) Sdf(x) dσd(x)− N f(xk)(cid:12)= hf,R(HC,QN;·)iKC ≤ kfkKC kR(HC,QN;·)kKC .
(cid:12)Z Xk=0 (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
In p(cid:12)articular, equality is assumed i(cid:12)n the last relation when taking f to be the ’representer’
( C,QN; ) itself. It follows that
R H ·
N−1
1
e(HC,QN)= kR(HC,QN;·)kKC = (cid:13) SdKC(·,x) dσd(x)− N KC(·,xk)(cid:13) . (8)
(cid:13)Z Xk=0 (cid:13)KC
(cid:13) (cid:13)
(cid:13) (cid:13)
Expandingthesquareoftheworst-caseerro(cid:13)randsubstitutingtheclosedformofthe(cid:13)reproducing
kernel we arrive at the well-known representation
[e( C,QN)]2 = ( C,QN; ), ( C,QN; )
H hR H · R H · i
N−1 N−1
2 1
= Sd SdKC(x,y) dσd(x) dσd(y)− N SdKC(x,xk) dσd(x)+ N2 KC(xk,xℓ)
Z Z k=0 Z k,ℓ=0
X X
5
N−1
1
= C x y dσ (x) dσ (y) x x . (9)
d Sd Sdk − k d d − N2 k k − ℓk
Z Z k,ℓ=0
X
 
Thisshowshow thesquareworst-case errorinourreproducingkernelHilbertspaceis related
to(2). Inthenextsectionweshowhowtheworst-caseerrore( C,QN)inourreproducingkernel
H
Hilbert space relates to the L spherical cap discrepancy.
2
4 Spherical cap discrepancy and Stolarsky’s invariance principle
Using the integral representation of the reproducing kernel (3) we have
1
KC(x,y) dσd(y)= 1C(z;t)(x)σd(C(z;t)) dσd(z) dt
ZSd Z−1ZSd
and
1 N−1 1 1 N−1
N k=0 KC(x,xk)= Z−1ZSd N k=0 1C(x;t)(z)1C(xk;t)(z) dσd(z) dt.
X X
Thus, the ’representer’ of the error of numerical integration is of the form (4); that is
1 1 N−1
R(HC,QN;x) = Z−1ZSd1C(z;t)(x)"σd(C(z;t))− N k=0 1C(xk;t)(z)#dσd(z) dt.
X
Therefore, using the inner product representation (5) in (8), we obtain
2 1/2
1 1 N−1
e( C,QN) = σd(C(z;t)) 1C(z;t)(xk) dσd(z) dt (10)
H Z−1ZSd(cid:12)(cid:12) − N Xk=0 (cid:12)(cid:12) 
(cid:12) (cid:12)
 (cid:12) (cid:12) 
statingthattheworst-caseerrorof(cid:12)thenumericalintegration formula(cid:12) QN in(1)intheconsidered
Sobolev space setting equals the so-called spherical cap L -discrepancy of the integration nodes.
2
Combining (9) and (10), we arrive at Stolarsky’s invariance principle for the Euclidean
distance on spheres.
Proposition 1 (Stolarsky [16]) Let x ,...,x Sd be an arbitrary N point configuration
0 N−1
∈
on the sphere Sd. Then we have
2
1 N−1 1 1 1 N−1
x x + σ (C(z;t)) 1 (x ) dσ (z) dt
N2 kX,ℓ=0k k − ℓk Cd Z−1ZSd(cid:12)(cid:12) d − N Xk=0 C(z;t) k (cid:12)(cid:12) d (11)
(cid:12) (cid:12)
(cid:12) (cid:12)
= x y dσd(x)(cid:12)dσd(y). (cid:12)
Sd Sdk − k
Z Z
The L -discrepancy of an N-point configuration on Sd decreases as its sum of distances
2
increases and vice versa. The right-hand side is the distance integral of the uniform measure σ
d
on the sphere Sd which is the uniqueextremal measure (also known as the equilibrium measure)
maximizing the distance integral
[µ]:= x y dµ(x)dµ(y)
I Sd Sdk − k
Z Z
over the family of (Borel) probability measures µ supported on Sd. For the potential theory of
the generalized distance integral we refer to Bjo¨rck [4].
6
5 A weighted reproducing kernel
The above results can be generalized by introducing a weight function. Let v : [ 1,1] R
− →
satisfy v(t) > 0 for all t and which has an antiderivative, which we denote by V. Then we define
the reproducing kernel with weight function v as follows
1
KC,v(x,y) = v(t) 1C(z;t)(x)1C(z;t)(y) dσd(z) dt, x,y Sd. (12)
Z−1 ZSd ∈
For functions represented by integrals
1
f (x) = g (z;t)1 (x) dσ (z) dt, i= 1,2,
i i C(z;t) d
Z−1ZSd
the corresponding inner product is now given by
1 1
f ,f = g (z;t)g (z;t) dσ (z) dt.
h 1 2iKC,v Z−1 v(t) ZSd 1 2 d
The reproducing kernel can be written as
KC,v(x,y) = V(min x,z , y,z ) dσd(z) V( 1), x,y Sd. (13)
Sd {h i h i} − − ∈
Z
For certain weight functionsv, this expressionmay havea conciseform. Thisreproducingkernel
defines a reproducing kernel Hilbert space C,v.
H
The ’representer’ of the error of numerical integration for the rule QN for functions in C,v
H
takes on the form
1 1 N−1
R(HC,v,QN;x) = Z−1ZSd1C(z;t)(x)v(t)"σd(C(z;t))− N k=0 1C(xk;t)(z)#dσd(z) dt.
X
We claim that KC,v(x,y) is a function of the inner product x,y , cf. Appendix B. Using the
h i
same approach as before we obtain
N−1
1
2
e(HC,v,QN) = N2 KC,v(xk,xℓ)− Sd SdKC,v(x,y) dσd(x) dσd(y). (14)
k,ℓ=0 Z Z
(cid:2) (cid:3) X
This worst case error can also be expressed in terms of a weighted discrepancy measure:
N−1
1
e(HC,v,QN)= (cid:13) SdKC,v(·,y) dσd(y)− N KC,v(·,xk)(cid:13) = R(HC,v,QN;·) KC,v
(cid:13)Z Xk=0 (cid:13)KC,v (cid:13) (cid:13)
(cid:13) (cid:13)
(cid:13) (cid:13) 2 (cid:13) 1/2 (cid:13)
(cid:13) 1 1 N−1 (cid:13)
= v(t) σ (C(x;t)) 1 (x ) dσ (x) dt . (15)
d C(x;t) k d
Z−1 ZSd(cid:12)(cid:12) − N Xk=0 (cid:12)(cid:12) 
(cid:12) (cid:12)
 (cid:12) (cid:12) 
Using (14) and (15) we obtain the weighted version of Stolarsky invariance principle.
(cid:12) (cid:12)
Theorem 1 Let x ,...,x Sd be an arbitrary N point configuration on the sphere Sd. Let
0 N−1
∈
KC,v be the weighted reproducing kernel given by (12). Then we have
2
1 N−1 1 1 N−1
N2 kX,ℓ=0KC,v(xk,xℓ)+Z−1v(t)ZSd(cid:12)(cid:12)σd(C(x;t))− N Xk=0 1C(x;t)(xk)(cid:12)(cid:12) dσd(x)dt
(cid:12) (cid:12)
(cid:12) (cid:12)
= KC,v(x,y)dσd(x)d(cid:12)σd(y). (cid:12)
Sd Sd
Z Z
7
The double integral above can be expressed in terms of the weight function, see (20).
In [5], Stolarsky’s (general) invariance principle is extended and used to get bounds for the
sphericalcapdiscrepancy, seealso[6]. Stolarksy[17]alsoextendedhisprincipletocertainmetric
spaces arising from measures.
Stolarsky [16] introduced the function
max{hx,yi,hy,ti}
ρ(x,y)= g(u)dudσ (z), x,y Sd,
d
ZSdZmin{hx,yi,hy,ti} ∈
which becomes a metric if the kernel g (integrable on [0,1]) is positive but the proof of the
corresponding invariance principle
2
1 N−1 1 1 N−1
ρ(x ,x )+2 v(t) σ (C(x;t)) 1 (x ) dσ (x) dt
N2 kX,ℓ=0 k ℓ Z−1 ZSd(cid:12)(cid:12) d − N Xk=0 C(x;t) k (cid:12)(cid:12) d (16)
(cid:12) (cid:12)
(cid:12) (cid:12)
= ρ(x,y) dσd(x) d(cid:12)σd(y) (cid:12)
Sd Sd
Z Z
making use of Haar integrals over the special orthogonal group SO(d+1) does not require it.
Note that for g 1 the function ρ(x,y) is a constant multiple of the Euclidean distance. With
≡
some care one may even consider g(x) = 1/(1 x2).
−
It is well-known that a reproducing kernel K(x,y) induces a distance (metric) by means of
d(x,y)= d (x,y)= K(x,x) 2K(x,y)+K(y,y).
K
−
p
For example, the reproducing kernel (7) yields
d (x,y) = C x y .
KC d k − k
In general, for the symmetric weighted kernelpKC,v(px,y), which does only depend on the inner
product x,y , it follows that (a Sd fixed)
h i ∈
1 1
2 2
KC,v(x,y) = dC,v(x,y) KC,v(x,x) KC,v(y,y) = dC,v(x,y) KC,v(a,a).
2 − − 2 −
n o
(cid:2) (cid:3) (cid:2) (cid:3)
By Theorem 1 on arrives at
2
1 N−1 1 1 N−1
2
N2 kX,ℓ=0(cid:2)dC,v(x,y)(cid:3) +2Z−1v(t)ZSd(cid:12)(cid:12)(cid:12)σd(C(x;t))− N Xk=0 1C(x;t)(xk)(cid:12)(cid:12)(cid:12) dσd(x)dt
2 (cid:12) (cid:12)
= dC,v(x,y) dσd(x)d(cid:12)σd(y), (cid:12)
Sd Sd
Z Z
(cid:2) (cid:3)
which should be compared with (16).
Acknowledgement: ThefirstauthorisgratefultotheSchoolofMathematicsandStatistics
at UNSW for their support.
A Auxiliary results
The normalized surface area measure σ on Sd admits the following decomposition
d
ω
dσ (y) = d−1 1 t2 d/2−1dtdσ (y∗), y = ( 1 t2y∗,t) Sd, (17)
d d−1
ω − − ∈
d
(cid:0) (cid:1) p
8
wheret [ 1,1], y∗ Sd−1 andω denotes thesurfaceareaofSd (cfMu¨ller[11]). (By definition
d
∈ − ∈
y,p = t, where p is the North Pole of Sd.) Thus, by rotational symmetry, the integral of a
hzonalifunction f( z,· ), z Sd fixed, with respect to σ reduces to
d
h i ∈
ω 1
f( z,y )dσ (y)= f( p,y )dσ (y) = d−1 f(t) 1 t2 d/2−1dt.
d d
ZSd h i ZSd h i ωd Z−1 −
(cid:0) (cid:1)
Proof. [Proof of relations (6)] One gets
1 1ω 1 1ω 1
C = p,y dσ (y) = d−1 t 1 t2 d/2−1dt = d−1 1 t2 d/2−12tdt
d d
2ZSd|h i| 2 ωd Z−1| | − 2 ωd Z0 −
1ω 1Γ((d+1)/2) (Bd) (cid:0) 1 (cid:1) (cid:0) (cid:1)
d−1 d
= = = H as d .
d ωd d √πΓ(d/2) d(Sd) ∼ √2πd → ∞
H
The second equality follows from
ω 1 ω 1 ω
1 = σ (Sd)= d−1 1 t2 d/2−1dt = d−1 v1/2−1(1 v)d/2−1dv = d−1 B(1/2,d/2),
d
ω − ω − ω
d Z−1 d Z0 d
(cid:0) (cid:1)
where B(a,b) = Γ(a)Γ(b)/Γ(a + b) is the beta function. The third equality follows from the
well-known formulas for the volume of the unit ball in Rd and the surface area of Sd. The
asymptotics follows from the asymptotic expansion of a ratio of Gamma functions (cf. [1]). 2
B The weighted reproducing kernel
Next, we investigate the weighted reproducing kernel (13) in more detail. In particular, it will
be shown that the kernel KC,v(x,y) is a function of the inner product x,y .
h i
On observing that x,z y,z if and only if y x,z 0 we may write for x = y
h i ≤ h i h − i ≥ 6
KC,v(x,y) = C,v(x,y)+ C,v(y,x) V( 1),
A A − −
which immediately shows symmetry of the reproducing kernel, where
y x
AC,v(x,y) = SdV(hx,zi)1[0,1]( y−x ,z )dσd(z). (18)
Z (cid:28)k − k (cid:29)
By abuse of notation we set (note that x,y =u)
h i
z = tx+ 1 t2z∗, 1 t 1,z∗ Sd−1,
− − ≤ ≤ ∈
y = ux+p1 u2y∗, 1 t 1,y∗ Sd−1.
− − ≤ ≤ ∈
p
In this way x will be the ’North Pole’ in the decomposition (17) and we obtain
ω 1 y x
AC,v(x,y) = ωd−d1 Z−1V(t)(cid:26)ZSd−11[0,1]((cid:28)ky−−xk,z(cid:29))σd−1(z∗)(cid:27) 1−t2 d/2−1dt.
(cid:0) (cid:1)
The indicator function in the inner integral is a zonal function depending on w = y∗,z∗ only.
h i
Thus, we apply again (17) with y∗ as ’North Pole’. That is
ω 1 ω 1 y x
AC,v(x,y) = ωd−1 V(t) ωd−2 1[0,1]( y−x ,z ) 1−w2 (d−1)/2−1dw 1−t2 d/2−1dt,
d Z−1 (cid:26) d−1 Z−1 (cid:28)k − k (cid:29) (cid:27)
(cid:0) (cid:1) (cid:0) (cid:1)
9
where the inner product evaluates as
y x 1+ x,y 1 x,y
− ,z = 1 t2 h iw −h it, x,y,z Sd,x = y. (19)
y x − 2 − 2 ∈ 6
(cid:28)k − k (cid:29) r r
p
Proceeding similarly for C,v(y,x), one sees that, indeed, C,v(y,x) = C,v(x,y). Further-
A A A
more, C,v(x,y) depends only on the inner product x,y which in turn implies that the
A h i
reproducing kernel KC,v(x,y) is a function of the inner product x,y .
h i
The right-hand side in (19) describes a line which stays strictly between the levels 1 and 1
−
for v in [ 1,1] by the left-hand side in (19). Further analysis gives that the indicator functions
−
is one (i) if t (1+u)/2 and 1 v 1, or, (ii) if (1+u)/2 t (1+u)/2 and
≤ − − ≤ ≤ − ≤ ≤
(t/√1 t2) (1 u)/(1+u) v 1, and zero otherwise. This leads to
− −p ≤ ≤ p p
p ω −√(1+u)/2
C,v(x,y) = d−1 V(t) 1 t2 d/2−1dt
A ω −
d Z−1
ω √(1+u)/2 (cid:0) (cid:1)
+ d−1 V(t)I ((d 1)/2,(d 1)/2) 1 t2 d/2−1dt,
(1−x(t))/2
ωd Z−√(1+u)/2 − − −
(cid:0) (cid:1)
where, when using u= x,y = cosφ (0 <φ < π) and t = cosψ, one has
h i
1+u 1 u 1 u t cotψ
= cos(φ/2), − = tan(φ/2), x(t)= − = .
2 1+u 1+u √1 t2 cot(φ/2)
r r r
−
The change of variable ξ =x(t) yields
ω −√(1+u)/2
C,v(x,y) = d−1 V(t) 1 t2 d/2−1dt
A ω −
d Z−1
ω 1 u d/2 1(cid:0) (cid:1)ξ dξ
d−1
+ − V( )I ((d 1)/2,(d 1)/2) ,
ωd (cid:18)1+u(cid:19) Z−1 1−u +ξ2 (1−ξ)/2 − − 1−u +ξ2 (d+1)/2
1+u 1+u
q (cid:16) (cid:17)
where we make use of the regularized incomplete beta function
z
I (a,b) = B (a,b)/B(a,b), B (a,b) = ta−1(1 t)b−1dt, a,b > 0.
z z z
−
Z0
We compute the following integral (using (18)):
y x
SdAC,v(x,y) dσd(y) = SdV(hx,zi) Sd1[0,1]( y−x ,z ) dσd(y) dσd(z).
Z Z Z (cid:28)k − k (cid:29)
The inner integral is one if z is in the half-sphere centered at x and zero otherwise. Hence,
−
ω 1
C,v(x,y) dσd(y) = d−1 V( t) 1 t2 d/2−1dt.
ZSdA ωd Z0 − −
(cid:0) (cid:1)
Since g′(t) = (ωd−1/ωd)(1 t2)d/2−1 for g(t) = (1/2)It2(1/2,d/2), 0 t 1, integration by
− ≤ ≤
parts gives
1 1 1
ZSdAC,v(x,y) dσd(y) = 2V(−1)+ 2 Z0 v(−t)It2(1/2,d/2)dt.
It follows that
ω 1
KC,v(x,y) dσd(x) dσd(y) = 2 d−1 V( t) 1 t2 d/2−1dt V( 1) (20a)
ZSdZSd ωd Z0 − − − −
1 (cid:0) (cid:1)
= v( t)It2(1/2,d/2)dt. (20b)
−
Z0
10