r 3n+2 F2n+1 φn+1 (r ) = (r ’ s)n+1 s n+1/2 Jn’1/2 (s)ds.

0

6.3 Radial functions 81

Thus by (6.9) we see that

r 3n+2 F2n+1 φn+1 (r ) = Bn f n (r ),

which is clearly nonnegative and nonvanishing. For an even space dimension d = 2n and

the same = n + 1 we only need to remark that φ 2n/2 +1 = φ (2n+1)/2 +1 . Hence φ 2n/2 +1

is positive de¬nite on R2n+1 and therefore also on R2n . The same argument proves the

positive de¬niteness of φ ( · 2 ) for > d/2 + 1.

The restriction ∈ N is actually not necessary but simpli¬es the proof. It is also possible

to allow real values for .

A general way of constructing positive de¬nite functions is to integrate a ¬xed posi-

tive de¬nite function against a nonnegative measure. That is exactly the way the Fourier

transform acts in Bochner™s characterization.

Theorem 6.21 Suppose that the continuous function φ : [0, ∞) ’ R is given by

∞

φ(r ) := (1 ’ r t)k’1 f (t)dt, (6.10)

+

0

where f ∈ C[0, ∞) is nonnegative and nonvanishing. Then φ is positive de¬nite on Rd if

k ≥ d/2 + 2.

Proof We ¬nd, for arbitrary coef¬cients and centers,

∞

N N N

± j ± φ( x j ’ x = ± j ± φk’1 (t x j ’ x f (t)dt ≥ 0

2) 2)

0

j=1 =1 j, =1

because φk’1 is positive de¬nite on Rd under the given assumptions, by Theorem 6.20.

Furthermore, since f is continuous, nonnegative, and nonvanishing the latter integral can

vanish only if ± ≡ 0.

A function of the form (6.10) obviously belongs to C k’2 , and the derivatives satisfy

(’1) φ ( ) (r ) ≥ 0 for 0 ¤ ¤ k ’ 2. Such functions are called multiply monotone functions.

De¬nition 6.22 Suppose that k ∈ N satis¬es k ≥ 2. A function φ : (0, ∞) ’ R is called

k-times monotone if (’1) φ ( ) is nonnegative, nonincreasing, and convex. If k = 1 we

require φ to be nonnegative and nonincreasing.

As in the case of Bochner™s theorem, the representation in (6.10) is not general enough to

characterize all k-times monotone functions. Again, the solution is to allow a more general

measure. Since this result will not play an important role for us, we just cite it for the

interested reader. The proof can be found in Williamson [201].

Theorem 6.23 A necessary and suf¬cient condition that φ : (0, ∞) ’ R is k-times mono-

tone is that φ is of the form

∞

φ(r ) = (1 ’ r t)+ dγ (t),

k’1

0

where γ is a nonnegative Borel measure.

82 Positive de¬nite functions

6.4 Functions, kernels, and other norms

Our investigation of positive de¬nite functions was motivated by the interpolation problem

in (6.2), using an interpolant of the form (6.1). This particular choice was very convenient

for our analysis, because we could restrict everything to the investigation of a single func-

tion . But of course (6.1) is not the only possible approach to the interpolation problem.

More generally, one would start with a function : Rd — Rd ’ C and try to form the

interpolant as

N

s f,X = ± j (·, x j ).

j=1

If we are interested only in sets of centers X = {x1 , . . . , x N } that are contained in a certain

subset ⊆ Rd , then it even suf¬ces to have a : — ’ C. To make the difference

from the previous approach clearer, we will call such a a kernel rather than a function.

De¬nition 6.24 A continuous kernel : — ’ C is called positive de¬nite on ⊆ Rd

if for all N ∈ N, all pairwise distinct X = {x1 , . . . , x N } ⊆ , and all ± ∈ C N \ {0} we have

N N

± j ±k (x j , xk ) > 0.

j=1 k=1

The de¬nition is in a certain way not precise. Since we have not speci¬ed the set it

might as well be a ¬nite set. In that situation it would be impossible to ¬nd for all N ∈ N

pairwise distinct points X . It should be clear that, in such a situation, only those N ∈ N

would have to be considered that allow the choice of N pairwise distinct data sites.

Radial basis functions ¬t into this more general setting by de¬ning (x, y) = φ( x ’ y 2 )

which leads to real-valued kernels. Most of the kernels we will discuss are radial, but we can

also use tensor products to create multivariate positive de¬nite functions from univariate

ones.

Proposition 6.25 Suppose that φ1 , . . . , φd are positive de¬nite and integrable functions on

R. Then

(x) := φ1 (x1 ) · · · φd (xd ), x = (x1 , . . . , xd )T ∈ Rd ,

is a positive de¬nite function on Rd .

Proof Since the univariate functions {φ j } are integrable, so also is the multivariate function

. Moreover, its d-variate Fourier transform is the product of the univariate Fourier

transforms:

(x) = φ1 (x1 ) · · · φd (xd ).

If we apply Theorem 6.11 to the univariate functions we see that their Fourier transforms

are nonnegative and nonvanishing. This means that the multivariate Fourier transform

6.4 Functions, kernels, and other norms 83

1

0.8

0.6

0.4

0.2

0

-1 -1

-0.5 -0.5

0 0

0.5 0.5

1 1

Fig. 6.3 The bivariate functions (x) = (1 ’ x 2 )2 (on the left) and (x) = (1 ’ |x1 |)+ (1 ’ |x2 |)+

+

(on the right).

also possesses this property. Hence, a ¬nal application of Theorem 6.11 shows that is

positive de¬nite.

In Figure 6.3 two compactly supported functions are shown. The function on the left hand

side is the radial function (x) = (1 ’ x 2 )2 , which is positive de¬nite on R2 by Theorem

+

6.20. The function on the right is the function (x) = (1 ’ |x1 |)+ (1 ’ |x2 |)+ , which is also

positive de¬nite on R2 by Theorem 6.20 and Proposition 6.25. We have chosen the smallest

possible exponent in both cases.

If functions : — ’ C are considered then one might naturally arrive at the ques-

tion whether there exists an extension of de¬ned on a bigger subset of Rd . Such a question

was discussed by Rudin in [159, 160] in the situation where the kernel is actually a func-

tion; to be more precise, where (x, y) = 0 (x ’ y) and 0 is a function that is de¬ned

on ’ = {x ’ y : x, y ∈ }. The results are diverse. On the one hand, it was shown by

Rudin [159] that if is a closed cube in Rd with d ≥ 2 then there always exists a positive

semi-de¬nite kernel of this form that does not have an extension to all of Rd . On the other

hand, Rudin [160] proved that every positive semi-de¬nite kernel of this form, de¬ned on a

ball and in addition radial, has an extension to Rd . In the case of a univariate setting, balls

and cubes are the same and hence an extension for functions that are positive semi-de¬nite

on intervals to the whole real line exists. Questions about uniqueness in this context were

considered in Akutowicz™ article [1].

One reason for looking at radial functions is that they allow easier computation. Hence,

one might think of investigating also p -radial functions, i.e. functions : Rd ’ R of the