and we will discuss in particular expansions of this form for the basis functions. But before

we do this we have to introduce another class of polynomials, which are important for

introducing the analogues of radial functions.

De¬nition 17.2 The (generalized) Legendre polynomial of degree in d ≥ 2 dimensions

is denoted by P = P (d; ·). It is normalized by P (d; 1) = 1 and satis¬es

ωd’1

1

P (t)Pk (t)(1 ’ t 2 )(d’3)/2 dt = δ ,k .

ωd’2 N (d, )

’1

There exists an intrinsic relation between generalized Legendre polynomials and spherical

harmonics.

310 Interpolation on spheres and other manifolds

Lemma 17.3 (Addition theorem) Between the spherical harmonics of order and the

Legendre polynomial of degree there exists the relation

N (d, )

N (d, )

Y ,k (x)Y ,k (y) = x, y ∈ S d’1 .

P (d; x T y),

ωd’1

k=1

We need another result concerning the asymptotic behavior of both the Legendre functions

and the spherical harmonics.

Lemma 17.4 Let Y ∈ π ⊥ (S d’1 ) = span{Y : 1 ¤ k ¤ N (d, )}; then

,k

N (d, )

|Y (x)| ¤ L 2 (S d’1 ) , x ∈ S d’1 .

Y

ωd’1

Moreover, the Legendre polynomials satisfy |P (d; t)| ¤ 1 for t ∈ [’1, 1].

17.2 Positive de¬nite functions on the sphere

In this section, we discuss and characterize positive de¬nite functions on the sphere. Of

course, the restriction of a positive de¬nite function on Rd to S d’1 forms a positive de¬nite

function on the sphere, and we will use this as a standard example. However, this does not

take the special situation of points on the sphere into account. Hence, we will investigate

positive de¬nite kernels of the form

∞ N (d, )

(x, y) = x, y ∈ S d’1 .

a ,k Y ,k (x)Y ,k (y), (17.2)

=0 k=1

As in the Rd case, radial kernels will play an important role. Note that radial now means

with respect to the geodesic distance.

: S d’1 — S d’1 is called radial or zonal if (x, y) =

De¬nition 17.5 A kernel

φ(dist(x, y)) = ψ(x T y) with univariate functions φ, ψ. The function ψ is called the shape

function of the kernel .

A ¬rst example of zonal functions comes from the Rd case. Suppose that = φ( · 2 ) :

Rd ’ R is a positive de¬nite and radial function on Rd . Since we have, for x, y ∈ S d’1 , that

x ’ y 2 = 2 ’ 2x T y, we can see that the restriction of to S d’1 has the representation

2

(x ’ y) = φ( x ’ y 2 ) = φ( 2 ’ 2x T y). Thus it is indeed a zonal function with shape

√

function ψ = φ( 2 ’ 2 ·). Note that the function φ here does√ coincide with the function

not

φ in De¬nition 17.5. That function is given by ψ —¦ cos = φ( 2 ’ 2 cos ·).

Zonal functions have the remarkable property that all the Fourier coef¬cients at a given

-level are the same.

=a ,1¤k¤

Proposition 17.6 A kernel of the form (17.2) is radial if and only if a ,k

N (d, ).

17.2 Positive de¬nite functions on the sphere 311

= a , 1 ¤ k ¤ N (d, ). Then by the addition theorem we have

Proof Suppose that a ,k

∞

a N (d, )

(x, y) = P (d; x T y),

ωd’1

=0

which shows that is radial. Conversely, if is radial then we can expand the shape

function ψ using the orthogonal basis P (d; ·) for L 2 [’1, 1] to get

∞

(x, y) = b P (d; x T y).

=0

The addition theorem and the uniqueness of the Fourier series give the rest.

Given an expansion of the form (17.2), it is natural to characterize a positive de¬nite

function by its Fourier coef¬cients. To allow point evaluations “ so far all expansions have

been in the L 2 -sense “ we have to assume that the coef¬cients decay fast enough. In the

case of a zonal kernel it obviously suf¬ces to require that

∞

|a |N (d, ) < ∞,

=0

since then

∞

|a |N (d, )

| (x, y)| ¤ |P (d; x T y)| < ∞,

ωd’1

=0

because of the bound on the Legendre polynomials from Lemma 17.4. The Weierstrass

M-test proves continuity. To state the corresponding assumption in the case of nonzonal

kernels we ¬rst de¬ne

|a ,k |

a := max (17.3)

1¤k¤N (d, )

and assume that

∞

a N (d, ) < ∞. (17.4)

=0

Since N (d, ) grows as O( d’2

) we see that (17.4) is satis¬ed if, for example,

’(d’1)’

a = O( ’ ∞,

), (17.5)

> 0.

with an

Lemma 17.7 Suppose that the coef¬cients of the kernel (17.2) satisfy (17.4); then is a

continuous function in both arguments.

312 Interpolation on spheres and other manifolds

N (d, )

Proof Let us ¬x y ∈ S d’1 and de¬ne S := a ,k Y ,k (y)Y ,k . The norm of this func-

k=1

tion satis¬es

N (d, ) N (d, )

N (d, )

= a 2,k |Y ,k (y)|2 ¤a |Y ,k (y)|2 = a 2 ,

2 2

S L 2 (S d’1 )

ωd’1

k=1 k=1