§1

⎪(’ 2 , ∞), ’1 < ± ¤ min 1 , » ’ 1 , if d = 1

⎪ 2 2

⎪

⎪

⎪[1, ∞),

⎪ ’ 1 < ± ¤ 1 », if d = 1

⎪

⎨ 2 2

» ∈ (’ 1 , ∞), ’1 < ± ¤ min 1 (» ’ 1 ), » ’ 1 , if d = 2,

⎪2 2 2 2

⎪

⎪

⎪

⎪[0, ∞), ’1 < ± ¤ 2 (» ’ 1), if d = 3,

1

⎪

⎪

©1

( 2 (d ’ 5), ∞), ’1 < ± ¤ 1 » ’ 1 (d ’ 1) , if d > 3.

2 2

Then the radial basis function (9.7) gives rise to a positive de¬nite function on Rd .

9.6 Notes and comments

Astonishingly, it needed quite some time for compactly supported radial basis functions to

be found. Everything started with the explicit construction of ˜Euclid™s hat™ in the present

author™s thesis [189], see also Schaback [162]; this is nothing other than the d-variate con-

volution of the characteristic function of the unit ball with itself. A little earlier, Narcowich

and Ward had used this function in [145] in a different context and without an explicit form.

The construction of the compactly supported functions of minimal degree was done

initially by the present author in [191] and partially published afterwards in [190, 192].

Nowadays, they are often simply called “Wendland™s functions”. A basis for these results

was given by the earlier publications by Chanysheva [40], Askey [6], and Gasper [68].

The operators I and D have become known to the radial basis function community

through Wu™s paper [203], but it seems that these operators have been known longer in the

¬eld of probability theory. In particular, Matheron [116] called them mont´ e and descent´ e.

e e

10

Native spaces

So far we have encountered positive de¬nite functions in the context of a scattered data

interpolation problem in Rd . In this chapter we want to take another point of view, which

also prepares us for the error analysis of the interpolation process. Our approach is motivated

by the following example. The Sobolev spaces on Rd can be de¬ned by

H s (Rd ) = { f ∈ L 2 (Rd ) : f (·)(1 + · ∈ L 2 (Rd )}.

2 s/2

2)

They can be equipped with an inner product

( f, g) H s (Rd ) := (2π )’d/2 f (ω)g(ω)(1 + ω 2 )s dω.

2

Rd

By the Sobolev embedding theorem it is well known that for s > d/2 the inclusion

H s (Rd ) ⊆ C(Rd ) holds, or, to be more precise, that every equivalence class in H s (Rd )

contains a continuous representer. We will always interpret H s (Rd ) as a set of continuous

functions in this way. A closer look at the inner product shows that it contains a nonnegative

weight function. In the case s > d/2 this weight function can be used to de¬ne a positive

de¬nite function by (ω) = (1 + ω 2 )’s . Actually, we know by Theorem 6.13 that

2

is given by

21’s s’d/2

(x) = x K d/2’s ( x 2 ).

2

(s)

Formal computations, which will be justi¬ed later on, give

f (ω) (ω)e’i x T ω

( f, (· ’ x)) H s (Rd ) = (2π )’d/2 dω = f (x).

(ω)

Rd

This reproducing property emphasizes the role of the function for the Sobolev space and

we will focus on it in the next section.

10.1 Reproducing-kernel Hilbert spaces

We are interested in vector spaces F consisting of functions f : ’ R de¬ned on a

region ⊆ Rd . The region can be quite arbitrary except that it should contain at least

one point. We consider only real vector spaces of real-valued functions. Very soon we will

133

134 Native spaces

see that on the one hand they correspond to real-valued positive semi-de¬nite kernels and

that on the other hand every real-valued positive de¬nite kernel leads naturally to a real

Hilbert space of real-valued functions. To include complex-valued kernels also, we would

have to discuss complex-valued function spaces. There are three reasons for not doing so.

First, both cases can be handled in a very similar way. The only difference is that in the

complex case special care has to be taken with the complex conjugate sign. Second, this

time there is no fundamental gain in a complex setting. While complex-valued functions

were indeed useful to derive Bochner™s and related results, the theory of reproducing-

kernel Hilbert spaces does not bene¬t from them. The main reason, however, is that all

relevant positive de¬nite functions are real-valued (because they are radial) and hence their

associated function spaces are also real spaces of real-valued functions. Nonetheless, we

will comment on the complex situation when appropriate and the capable reader will have

no problem in stating and proving the corresponding results.

De¬nition 10.1 Let F be a real Hilbert space of functions f : ’ R. A function :

— ’ R is called a reproducing kernel for F if

(1) (·, y) ∈ F for all y ∈ ,

(2) f (y) = ( f, (·, y))F for all f ∈ F and all y ∈ .

The reproducing kernel of a Hilbert space is uniquely determined. Suppose there are two

reproducing kernels 1 and 2 . Then property (2) gives ( f, 1 (·, y) ’ 2 (·, y))F = 0 for

all f ∈ F and all y ∈ . Setting f = 1 (·, y) ’ 2 (·, y) for a ¬xed y shows the uniqueness.

Let us give a ¬rst characterization of a Hilbert function space with a reproducing kernel.

Theorem 10.2 Suppose that F is a Hilbert space of functions f : ’ R. Then the fol-

lowing statements are equivalent:

(1) the point evaluation functionals are continuous, i.e. δ y ∈ F — for all y ∈ ;

(2) F has a reproducing kernel.

Proof Suppose that the point evaluation functionals are continuous. By Riesz™ representa-

tion theorem we ¬nd, for every y ∈ , a y ∈ F such that δ y ( f ) = ( f, y )F for all f ∈ F.

Thus, (x, y) := y (x) is the reproducing kernel of F. Now suppose that F has a repro-

ducing kernel . This means that δ y = (·, (·, y))F for y ∈ . Since the inner product is

continuous, so is δ y .

A reproducing-kernel Hilbert space has many special features. We collect some of them

now.

Theorem 10.3 Suppose F is a Hilbert space of functions f : ’ R with reproducing

kernel . Then we have

(1) (x, y) = ( (·, x), (·, y))F = (δx , δ y )F — for x, y ∈ ,

(2) (x, y) = (y, x) for x, y ∈ ,

(3) if f, f n ∈ F, n ∈ N, are given such that f n converges to f in the Hilbert space norm then f n also

converges pointwise to f .

10.1 Reproducing-kernel Hilbert spaces 135

Proof The Riesz™ representation F : F — ’ F reduces for point evaluations to F(δ y ) =

(·, y) because of the reproducing-kernel properties. This means that

(δx , δ y )F — = (F(δx ), F(δ y ))F = ( (·, x), (·, y))F .

Furthermore,

(x, y) = δx ( (·, y)) = ( (·, y), (·, x))F = ( (·, x), (·, y))F .

Hence, property (1) is proven. Property (2) follows immediately from Property (1). Property

(3) is a consequence of

| f n (x) ’ f (x)| = |( f n ’ f, (·, x))F | ¤ f n ’ f F.