N (d, ) N (d, )

|S (x)| ¤ ¤a , x ∈ S d’1 ,

S L 2 (S d’1 )

ωd’1 ωd’1

so that

∞ ∞

N (d, )

| (x, y)| ¤ |S (x)| ¤ <∞

a

ωd’1

=0 =0

by assumption (17.4). Continuity now follows by the Weierstrass M-test again.

Next, we come to the problem of ¬nding positive de¬nite kernels. Remember that a

positive de¬nite kernel is by de¬nition continuous and symmetric. Because of the results in

the Rd case the following theorem should not be a surprise. Since the kernel is symmetric

and real-valued we can restrict ourselves to real coef¬cient vectors in the quadratic form.

Theorem 17.8 Suppose that the kernel (17.2) is continuous. Then it is positive semi-de¬nite

if and only if all coef¬cients are nonnegative. Moreover, if all coef¬cients are positive then

is positive de¬nite.

Proof For given pairwise distinct points X = {x1 , . . . , x N } ⊆ S d’1 and a vector ± ∈ R N ,

we can express the quadratic form as follows:

2

∞ N (d, )

N N

±i ± j (xi , x j ) = ± j Y ,k (x j ) .

a ,k

=0 k=1

i, j=1 j=1

Hence, if all coef¬cients are nonnegative then clearly the quadratic form is nonnegative.

Moreover, if the quadratic form vanishes and if all coef¬cients are positive we must have

N

± j Y (x j ) = 0,

j=1

for every spherical polynomial Y . Since N is ¬nite, we can ¬nd for each 1 ¤ j ¤ N a

spherical polynomial Y j with Y j (xi ) = δi j , which shows that ± j = 0.

It remains to demonstrate that a positive semi-de¬nite function has nonnegative Fourier

coef¬cients. The easiest way to do this is to use an equivalent characterization of positive

semi-de¬nite functions, namely integrally positive semi-de¬nite functions. Even though we

cannot apply Proposition 6.4 directly, its proof implies that

γ (x)γ (y) (x, y)d S(x)d S(y) ≥ 0

S d’1 S d’1

17.2 Positive de¬nite functions on the sphere 313

for all γ ∈ C(S d’1 ). Inserting the expansion (17.2) for and setting γ = Y»,κ shows that

a»,κ ≥ 0.

In the case of radial functions this reduces to the following result by Schoenberg [174].

(x, y) = φ(dist(x, y)), x, y ∈ S d’1 , is

Corollary 17.9 (Schoenberg) A radial function

positive semi-de¬nite if and only if

∞

φ(r ) = b P (d; cos r )

=0

with

∞

b ≥ 0 for all ∈ N0 b < ∞.

and (17.6)

=0

Moreover, if all coef¬cients b are positive then is positive de¬nite.

is radial, its shape function ψ has the representation ψ(r ) =

Proof Since

∞

b P (d; r ). The coef¬cients are given by b = a N (d, )/ωd’1 . Hence, from The-

=0

orem 17.8 we have immediately that on the one hand (17.6) implies that is positive

semi-de¬nite and that is positive de¬nite if all coef¬cients are positive.

On the other hand, in the case of a positive semi-de¬nite kernel, Theorem 17.8 shows that

all coef¬cients b have to be nonnegative. But then, since is continuous, we can conclude

that φ(0) = ∞ b < ∞.

=0

Considering radial or zonal basis functions, one can start with a univariate function φ

and ask the question whether it is positive de¬nite on every sphere S d’1 . Such functions

must exist because every radial function that is positive de¬nite on every Rd is one of them.

Moreover, the function φ(r ) = cos r is positive semi-de¬nite on every sphere S d’1 since

the quadratic form becomes simply

2

N N N

± j ±k φ(dist(x j , xk )) = ± j ±k x T x k = ±jxj ≥ 0.

j

j,k=1 j,k=1 j=1 2

Furthermore, the product of two positive semi-de¬nite functions is in turn positive semi-

de¬nite and the same is true if we form linear combinations with nonnegative coef¬cients.

This establishes the suf¬cient part of the next result. For the necessary part, we refer the

interested reader again to Schoenberg™s paper [174].

Theorem 17.10 (Schoenberg) A function φ is positive de¬nite on every sphere S d’1 , d ≥ 2,

if it has the representation

∞

φ(r ) = b cos r,

=0

b < ∞.

with nonnegative coef¬cients b that satisfy

314 Interpolation on spheres and other manifolds

17.3 Error estimates

The investigation of the interpolation error follows the lines of the Rd case. This means that

we employ the native Hilbert space N (S d’1 ) for a positive de¬nite kernel ∈ C(S d’1

— S d’1 ) and bound the interpolation error f ’ s f,X for a function f ∈ N (S d’1 ) in terms

of the ¬ll distance.

Throughout this section we will assume that the Fourier coef¬cients a ,k of the kernel

are positive and that they satisfy the decay condition (17.4). From our general theory

on native spaces it should be clear what they are in this case. In particular, Theorem 10.29

gives the characterization

∞ N (d, ) ∞ N (d, )

| f ,k |2

N (S )= f= <∞ ,

d’1

f ,k Y ,k :

a ,k

=0 k=1 =0 k=1

and the inner product takes the form

∞ N (d, )

f ,k g ,k

= .

( f, g)N (S d’1 )

a ,k

=0 k=1

To derive error estimates, we split the error into a product of two terms, the power function

and the native space norm of f , as we did in Theorem 11.4. The condition that is open was

only necessary for deriving estimates on the derivatives. Here, we want to restrict ourselves

to the case of a pure interpolation error, so we do not need to incorporate this condition.

The next step is to bound the power function in terms of the ¬ll distance, which now

takes the form

h X := sup inf dist(x, x j ),

x∈S d’1 x j ∈X

paying tribute to the special topology of the sphere. For h X we have dropped the additional

index that indicated the domain, since we are now just working on the sphere.

The original idea behind bounding the power function was to use its minimization property

with respect to its coef¬cients and to construct a local polynomial reproduction. The same is

possible here, but the compactness of the sphere makes the locality unnecessary. Hence, we