ńņš. 38 |

quences. His work essentially contained the characterization by Bernstein. But Bernstein

was obviously not aware of the work of Hausdorff and gave an independent proof. Later,

in 1931, Widder gave another independent proof [199].

The ļ¬rst two sections of this chapter are based on Widderā™s work, as it is represented in

his book [200] but employing a measure-theoretical approach rather than Laplaceā“Stieltjes

integrals.

Schoenberg proved his characterization (Theorem 7.13) in 1938 [173] by considering

Bochnerā™s characterization via Fourier transforms in the case of radial functions. He inves-

tigated what happens if the space dimension tends to inļ¬nity.

The approach taken here seems to be more elementary and, from a certain point of view,

also more elegant. It originates from work by Wells and Williams [188] and Kuelbs [99].

8

Conditionally positive deļ¬nite functions

The interpolation problem (6.2) led us to the idea of using positive deļ¬nite functions.

But not all popular choices of radial basis functions that are used ļ¬t into this scheme.

The thin-plate spline may serve us as an example. Suppose the basis function is given

by (x) = x 2 log( x 2 ), x ā Rd . Let N = d + 1 and let the centers be the vertices of

2

a regular simplex whose edges are all of unit length. Then all entries (x j ā’ xk ) of the

interpolation matrix are zero.

In this chapter we generalize the notion of positive deļ¬nite functions in a way that covers

all the relevant possibilities for basis functions. We will derive characterizations that can be

seen as the generalizations of Bochnerā™s and Schoenbergā™s results.

8.1 Deļ¬nition and basic properties

As in the case of positive deļ¬nite functions we distinguish between real-valued and complex-

valued functions, but we now have to be more careful.

Deļ¬nition 8.1 A continuous function : Rd ā’ C is said to be conditionally positive semi-

deļ¬nite of order m (i.e. to have conditional positive deļ¬niteness of order m) if, for all N ā N,

all pairwise distinct centers x1 , . . . , x N ā Rd , and all Ī± ā C N satisfying

N

Ī± j p(x j ) = 0 (8.1)

j=1

for all complex-valued polynomials of degree less than m, the quadratic form

N

Ī± j Ī±k (x j ā’ xk ) (8.2)

j,k=1

is nonnegative. is said to be conditionally positive deļ¬nite of order m if the quadratic

form is positive, unless Ī± is zero.

A ļ¬rst important fact on conditionally positive (semi-)deļ¬nite functions concerns their

order.

97

98 Conditionally positive deļ¬nite functions

Proposition 8.2 A function that is conditionally positive (semi-)deļ¬nite of order m is also

conditionally positive (semi-)deļ¬nite of order ā„ m. A function that is conditionally positive

(semi-)deļ¬nite of order m on Rd is also conditionally positive (semi-)deļ¬nite of order m on

Rn with n ā¤ d.

This means for example that every positive deļ¬nite function has also conditional positive

deļ¬niteness of any order.

Another consequence is that it is natural to look for the smallest possible order m. Hence,

when speaking of a conditionally positive deļ¬nite function of order m we give in general

the minimal possible m.

As in the case of positive deļ¬nite functions the deļ¬nition reduces to real coefļ¬cients

and polynomials if the basis function is real-valued and even. This is in particular the case

whenever is radial.

Theorem 8.3 A continuous, even function : Rd ā’ R is conditionally positive deļ¬nite of

order m if and only if, for all N ā N, all pairwise distinct centers x1 , . . . , x N ā Rd , and all

Ī± ā R N \ {0} satisfying

N

Ī± j p(x j ) = 0

j=1

for all real-valued polynomials of degree less than m, the quadratic form

N

Ī± j Ī±k (x j ā’ xk )

j,k=1

is positive.

We cannot conclude from the deļ¬nition of a conditionally positive semi-deļ¬nite function

alone that (x) = (ā’x) is automatically satisļ¬ed, as was the case for positive semi-

deļ¬nite functions. This is a consequence of the following proposition.

Proposition 8.4 Every polynomial q of degree less than 2m is conditionally positive semi-

deļ¬nite of order m. More precisely, for all sets {x1 , . . . , x N } ā Rd and all Ī± ā C N satisfying

(8.1) for all p ā Ļmā’1 (Rd ), the quadratic form (8.2) for = q is identically zero.

cĪ² x Ī² we can use the multino-

Proof With multi-indices Ī², Īŗ ā Nd and q(x) = |Ī²|<2m

0

mial theorem to derive

N N

Ī± j Ī±k cĪ² (x j ā’ xk )Ī²

Ī± j Ī±k q(x j ā’ xk ) =

|Ī²|<2m j,k=1

j,k=1

Ī² N N

Ī²ā’Īŗ

(ā’1)|Īŗ| Īŗ

= Ī±jxj Ī±k x k .

cĪ²

Īŗ

Īŗā¤Ī²

|Ī²|<2m j=1 k=1

Since |Ī²| < 2m it is impossible that both |Īŗ| ā„ m and |Ī² ā’ Īŗ| ā„ m. Thus either the sum

Ī²ā’Īŗ Īŗ

N N

j=1 Ī± j x j or the sum k=1 Ī±k xk must vanish due to (8.1) for all pairs Ī², Īŗ.

8.1 Deļ¬nition and basic properties 99

It can be shown that every polynomial with a degree greater than 2m cannot be condi-

tionally positive semi-deļ¬nite of order m (cf. Sun [182]).

The conditional positive deļ¬niteness of order m of a function can also be interpreted

as the positive deļ¬niteness of the matrix A ,X = ( (x j ā’ xk )) on the space of vectors Ī±

such that

N

Ī± j p (x j ) = 0, 1ā¤ ā¤ Q = dim Ļmā’1 (Rd ).

j=1

Thus, in this sense, A ,X is positive deļ¬nite on the space of vectors Ī± āperpendicularā to

polynomials. Let us dwell on this subject a little more. Each pair consisting of a vector

Ī± ā C N and a set of pairwise distinct points X = {x1 , . . . , x N } that together satisfy (8.1)

for all polynomials of degree less than m deļ¬ne a linear functional

N

Ī»Ī±,X := Ī± j Ī“x j ,

j=1

where Ī“x denotes point evaluation at x. Deļ¬ne Ļmā’1 (Rd )ā„ to be the space of all such

functionals. Then Ī± is admissible in the deļ¬nition of a conditionally positive semi-deļ¬nite

function if and only if Ī»Ī±,X ā Ļmā’1 (Rd )ā„ .

The case m = 1, which also appears in the linear algebra literature, is usually dealt with

using ā¤ and is then referred to as (conditionally or almost) negative deļ¬nite. In this case

the constraint on the Ī± j is simply N Ī± j = 0.

j=1

Since the matrix A ,X is conditionally positive deļ¬nite of order m, it is positive deļ¬nite on

a subspace of dimension N ā’ Q, Q = dim Ļmā’1 (Rd ). Thus it has the interesting property

that at least N ā’ Q of its eigenvalues are positive. This follows immediately from the

Courantā“Fischer theorem. In the case m = 1 we can make an even stronger statement.

Theorem 8.5 Suppose that is conditionally positive deļ¬nite of order 1 and that (0) ā¤ 0.

Then the matrix A ,X ā R N Ć—N has one negative and N ā’ 1 positive eigenvalues and in

particular it is invertible.

Proof From the Courantā“Fischer theorem we conclude that A ,X has at least N ā’ 1 pos-

n

itive eigenvalues. But since 0 ā„ N (0) = tr(A ,X ) = i=1 Ī»i , where the Ī»i denote the

eigenvalues of A ,X and tr(A ,X ) its trace, A ,X must also have at least one negative

eigenvalue.

ńņš. 38 |