ńņš. 33 |

of radial basis function interpolation; hence we do not want to discuss them in much detail.

Nonetheless, there might be some applications that would beneļ¬t from basis functions of

this particular form. Thus we will give at least a certain amount of information on this

challenging topic. The ļ¬rst thing is that the situation is similar to the 2 case if the 1 -norm

is used. In this situation it is also possible to characterize all positive semi-deļ¬nite functions.

Theorem 6.26 A function : Rd ā’ R deļ¬ned by (x) = Ļ( x 1 ), x ā Rd , is positive

semi-deļ¬nite if and only if there exists a ļ¬nite Borel measure Ī± on [0, ā) such that

ā

Ļ(r ) = Ļ0 (r t)dĪ±(t),

0

84 Positive deļ¬nite functions

where Ļ0 (r ) is given by

ā

2d/2 2 (d/2)

r ā’(dā’2)/2 (t 2 ā’ 1)(dā’3)/2 t ā’(3dā’4)/2 J(dā’2)/2 (r t)dt.

Ļ0 (r ) = ā

Ļ ((d ā’ 1)/2) 1

A proof of this result was given by Cambanis et al. [36]. While it characterizes positive

semi-deļ¬nite functions that are 1 -radial, things are worse for p -radial functions when

p > 2. In this case it can be shown that for space dimension d ā„ 3 the only function

(x) = Ļ( x p ) that is positive semi-deļ¬nite is the trivial function ā” 0. Nonetheless,

there is some hope for space dimensions d ā¤ 2 and for p ā [1, 2]. Details on the negative

result can be found in Zastavnyi [209].

6.5 Notes and comments

Positive (semi-)deļ¬nite functions play an important role not only in approximation theory

but also in other mathematical areas, for example in probability theory. There, a positive

(semi-)deļ¬nite function is nothing other than the characteristic function of a probability

distribution. Hence, the basic properties listed at the beginning of this chapter are folklore

nowadays and the reader might ļ¬nd more information in the review article by Stewart [181]

or the book by Lukacs [110]. But in contrast with probability theory, where semi-deļ¬nite

functions work as well as deļ¬nite ones, approximation theory has to stick with positive

deļ¬nite functions, because being positive deļ¬nite is crucial for interpolation.

The most important result of this chapter is without a question Bochnerā™s theorem

(Theorem 6.6). But even though nowadays all tribute goes to Bochner for this result, Mathias

[117] had proved already, in 1923, that a univariate positive (semi-)deļ¬nite function has a

nonnegative Fourier transform. The proof of Bochnerā™s theorem, given here in the modern

language of measure theory, was motivated by Donoghueā™s presentation [46].

The truncated power function in Theorem 6.20 has been investigated by several authors,

for example Askey [6] and Chanysheva [40]. It will play an important role in what follows.

7

Completely monotone functions

At the end of the last chapter we discussed radial positive deļ¬nite functions. We investigated

whether a univariate function Ļ : [0, ā) ā’ R deļ¬nes a positive deļ¬nite function (x) =

Ļ( x 2 ) on a ļ¬xed Rd . But we already had examples where the univariate function gives

rise to a positive deļ¬nite function on every Rd . This is of course a very pleasant feature.

The reader should reļ¬‚ect on this property for a moment. It means that we can use the same

univariate function Ļ to interpolate any number of scattered data in any space dimension.

Hence, we now want to discuss such functions in greater detail.

In the last chapter, we also encountered k-times monotone functions and noticed their

connection with positive deļ¬nite functions on Rd where k and d were related in a certain

way.

It will turn out that there is a similar connection between completely monotone functions,

which are the generalization of multiply monotone functions, and radial functions that are

positive deļ¬nite on every Rd . To be more precise, suppose Ļ : [0, ā) ā’ R is given by

ā

eā’tr dĪ½(t),

2

Ļ(r ) = (7.1)

0

with a nonnegative and ļ¬nite Borel measure Ī½ deļ¬ned on the Borel sets B([0, ā)) =

[0, ā) ā© B(R). Then, for arbitrary x1 , . . . , x N and an arbitrary Ī± ā R N , we have

ā

N N

Ī± j Ī±k eā’t x j ā’xk 2

Ī± j Ī±k Ļ( x j ā’ xk 2 ) = dĪ½(t) ā„ 0, (7.2)

2

0

j,k=1 j,k=1

because the Gaussians involved are positive deļ¬nite and the measure Ī½ is nonnegative.

Moreover, Ļ is continuous because Ī½ is ļ¬nite. This means that Ļ is positive semi-deļ¬nite

ā

on every Rd . Furthermore, obviously f (r ) := Ļ( r ) satisļ¬es

ā ā

dn n

nd ā’r t

t n eā’r t dĪ½(t) ā„ 0.

f (r ) = (ā’1) dĪ½(t) =

n

(ā’1) e

dr n dr n 0 0

Differentiation under the integral sign is justiļ¬ed because the measure Ī½ is ļ¬nite. Hence,

if Ļ is a function of the form (7.1) then it is positive semi-deļ¬nite on every Rd and the

ā

associated function f = Ļ( Ā·) satisļ¬es (ā’1)n f (n) ā„ 0. The goal of this chapter is to prove

that all three properties are actually equivalent.

85

86 Completely monotone functions

7.1 Deļ¬nition and ļ¬rst characterization

We start by deļ¬ning completely monotone functions.

Deļ¬nition 7.1 A function Ļ is called completely monotone on (0, ā) if it satisļ¬es Ļ ā

C ā (0, ā) and

(ā’1) Ļ (r ) ā„ 0

for all ā N0 and all r > 0. The function Ļ is called completely monotone on [0, ā) if it

is in addition in C[0, ā).

The ļ¬rst equivalent characterization of completely monotone functions can be derived by

using iterated forward differences.

Deļ¬nition 7.2 Let k ā N0 . Suppose that { f j } jāN0 is a sequence of real numbers. The kth-

order iterated forward difference is

k

k

{ f j }( ) ā” +j, ā N0 .

k k

(ā’1)kā’ j

f := f (7.3)

j

j=0

For a function Ļ : [0, ā) ā’ R we deļ¬ne the kth-order difference by

k

k

h Ļ(r ) Ļ(r + j h),

k

(ā’1)kā’ j

:= (7.4)

j

j=0

for any r ā„ 0 and h > 0. If Ļ is deļ¬ned only on (0, ā) then we restrict r in (7.4) to r > 0.

Obviously, for a ļ¬xed r ā„ 0 and a ļ¬xed h > 0, we have h Ļ(r ) = { f j }(0) with the

k k

sequence { f j } given by f j := Ļ(r + j h).

Lemma 7.3 Suppose that Ļ : (0, ā) ā’ R satisļ¬es (ā’1)n n Ļ(r ) ā„ 0 for all r, h > 0

h

and n = 0, 1, 2. Then Ļ is nonnegative, nonincreasing, continuous, and convex on

(0, ā).

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