8.6 Notes and comments

One might say that the whole radial basis function theory started with the practical work of

Hardy in 1971 on multiquadrics (see [79] and also the review article [80]) and with the the-

oretical work of Duchon on thin-plate splines (see [47“49]) in the late 1970s. Shortly there-

after, Meinguet [122“124] popularized thin-plate splines as a practical numerical method

for multivariate interpolation. In the mid 1980s, Micchelli [133] put Hardy™s multiquadrics

on a ¬rm mathematical basis by solving a conjecture, drawn from computational experience,

118 Conditionally positive de¬nite functions

of Franke by connecting it with the classical results of Bochner and Schoenberg on positive

de¬nite functions and thereby releasing it from the limitations of the variational perspectives

of thin-plate splines and the specialized form of multiquadrics.

The proof of Micchelli™s theorem, which consists in its original form of only the suf¬cient

part, was completed in [77] by Guo et al. seven years later in an even stronger version. As

for positive de¬nite and radial functions, it is not necessary to assume φ to be in C ∞ (0, ∞).

This can be concluded from the fact that φ( · 2 ) is conditionally positive de¬nite on every

2

Rd . The simple proof given here was based upon Sun™s paper [183].

Several results on conditionally positive de¬nite functions employ distribution theory,

pseudo-functions, or both; see for example Madych and Nelson [113] and Gel™fand and

Vilenkin [69]. The approach here tries to avoid the use of such tools even if it sometimes

recovers the ideas behind them. In any case, to introduce straightforwardly the concept that

the generalized Fourier transform of a function with a possible singularity at the origin is

a function itself seems to be a major step in simplifying this theory. It came ¬rst up in the

unpublished preprint [111] by Madych and Nelson. Some of the examples of generalized

Fourier transforms can be found in the books by Gel™fand and Vilenkin [69] and by Jones

[95].

9

Compactly supported functions

In numerical analysis, the concept of locally supported basis functions is of general im-

portance. Several function spaces used for approximation possess locally supported basis

functions. The most prominent examples in the one-dimensional case are the well-known

B-splines. The general advantages of compactly supported basis functions are a sparse in-

terpolation matrix on the one hand, and the possibility of a fast evaluation of the interpolant

on the other.

Thus, it seems to be natural to look for locally supported functions also in the context

of radial basis function interpolation and we will give an introduction to this ¬eld in this

chapter.

At the outset, though, we want to point out one crucial difference from classical spline

theory. While the support radius of the B-splines can be chosen proportional to the max-

imal distance between two neighboring centers, something similar will not lead to a con-

vergent scheme in the theory of radial basis functions. The correct choice of the support

radius is a very delicate question, which we will address in a later chapter on numerical

methods.

9.1 General remarks

Gaussians, (inverse) multiquadrics, powers, and thin-plate splines share two joint features.

They are all globally supported and are positive de¬nite on every Rd . The truncated powers

from Theorem 6.20, however, are compactly supported but are also restricted to a ¬nite

number of space dimensions.

We will see that the two features are connected. But let us ¬rst comment on conditionally

positive de¬nite functions.

Theorem 9.1 Assume that the function : Rd ’ C is continuous and compactly sup-

ported. If is conditionally positive de¬nite of minimal order m ∈ N0 then m is necessarily

zero, i.e. is must be positive de¬nite.

Proof Since is integrable, it possesses a classical Fourier transform that is continuous.

In this situation the generalized Fourier transform coincides with the classical one. Hence,

119

120 Compactly supported functions

by Theorem 8.12 the Fourier transform is nonnegative in Rd \ {0} and not identically zero.

Since it is continuous we also have (0) ≥ 0, and Theorem 6.11 ensures that is positive

de¬nite.

Thus we can concentrate on positive de¬nite radial functions with compact support and

use the classical Fourier transform instead of the generalized Fourier transform to handle

them.

The next theorem shows that the Fourier transform is indeed the right tool to handle such

functions, not the Laplace transform, which we have seen to be important in the context of

completely monotone functions.

Theorem 9.2 Suppose the continuous and nonvanishing function φ : [0, ∞) ’ R is posi-

tive de¬nite on every Rd . Then φ(r ) = 0 for all r ∈ [0, ∞).

Proof Since φ is positive de¬nite on every Rd there exists a ¬nite nonnegative Borel

measure μ on [0, ∞) such that

∞

e’r u dμ(u).

2

φ(r ) =

0

If φ had a zero r0 , this would mean that

∞

e’r0 u dμ(u).

2

0=

0

As e’r0 u > 0 for all u ≥ 0, we must have μ([0, ∞)) = 0 and hence φ ≡ 0, which contradicts

2

the fact that φ is nonvanishing.

An immediate consequence of the preceding theorem is that the dimensions d, on which

a compactly supported φ is positive de¬nite, are restricted to a ¬nite number. If φ is not

positive de¬nite on a ¬xed Rd0 then it cannot be positive de¬nite on any higher-dimensional

space.

Corollary 9.3 A continuous, univariate, and compactly supported function φ cannot be

positive de¬nite on every Rd .

9.2 Dimension walk

From the results of the last section we know that if we want to construct locally supported,

radial, and positive de¬nite functions we have to work with a ¬xed space dimension d. In

this case the Fourier transform is the right tool. Following Bochner, we know that a positive

de¬nite function on Rd is characterized by a nonnegative d-variate Fourier transform. In the

case of a radial function = φ( · 2 ) ∈ L 1 (Rd ) this is a radial function = Fd φ( · 2 )

again (see Theorem 5.26), where

∞

Fd φ(r ) = r ’(d’2)/2 φ(t)t d/2 J(d’2)/2 (r t) dt.

0

9.2 Dimension walk 121

This operator Fd , which acts on univariate functions, can be manipulated by operators that

we now want to introduce.

De¬nition 9.4

(1) Let φ be given such that t ’ φ(t)t is in L 1 [0, ∞); then we de¬ne for r ≥ 0

∞

(Iφ)(r ) = tφ(t)dt.

r

(2) For even φ ∈ C 2 (R) we de¬ne for r ≥ 0

1

(Dφ)(r ) = ’ φ (r ).

r

In both cases the resulting functions should be seen as even functions by even extension.

Thus I and D map even univariate functions to even univariate functions by even exten-

sion. Both operators respect a compact support.

Note that the function Dφ is continuous at zero. Since φ ∈ C 2 (R) is even we have φ (t) =

’φ (’t) and in particular φ (0) = 0. This means that φ (t) = O(t) for t ’ 0 and hence

Dφ(t) = O(1) for t ’ 0. Moreover, the operators I and D are inverse in the following sense.

Lemma 9.5 If φ is continuous and satis¬es t ’ tφ(t) ∈ L 1 [0, ∞) then DIφ = φ. Con-

versely, if φ ∈ C 2 (R) is even and satis¬es φ ∈ L 1 [0, ∞) then IDφ = φ.