The interaction between the operators I, D and Fd is given by the next theorem. Re-

member that t ’ φ(t)t d’1 ∈ L 1 [0, ∞) implies in particular that = φ( · 2 ) ∈ L 1 (Rd ).

Theorem 9.6 Suppose that φ is continuous.

(1) If t ’ φ(t)t d’1 ∈ L 1 [0, ∞) and d ≥ 3 then Fd (φ) = Fd’2 (Iφ).

(2) If φ ∈ C 2 (R) is even and t ’ φ (t)t d ∈ L 1 [0, ∞) then Fd (φ) = Fd+2 (Dφ).

Proof To prove the ¬rst statement, we start by showing that the function r ’ Iφ(r )r d’3 ∈

L 1 [0, ∞) and hence that Iφ( · 2 ) ∈ L 1 (Rd’2 ). Since d ≥ 3 the function Iφ is well de¬ned

and continuous. Moreover, for R > 0 we ¬nd that

∞

R R

|Iφ(r )|r d’3 dr ¤ |φ(t)|tr d’3 dtdr

0 0 r

∞

R R R

= |φ(t)|tr d’3 dtdr + |φ(t)|tr d’3 dtdr

0 r 0 R

and we have to bound each of the last two integrals uniformly in R. For the ¬rst of these

we exchange the order of integration to get

R R R t

|φ(t)|tr d’3 dtdr = |φ(t)|tr d’3 dr dt

0 r 0 0

R

1

= t d’1 |φ(t)|dt,

d ’2 0

122 Compactly supported functions

which is obviously bounded by t d’1 φ(t) L 1 [0,∞) /(d ’ 2). The second integral is actually

a product of two univariate integrals and allows the bound

∞ ∞

R

R d’2

t d’1 |φ(t)|t ’d+2 dt

|φ(t)|tr dtdr =

d’3

d ’2

0 R R

∞

1

¤ t d’1 |φ(t)|dt

d ’2 R

∞

1

¤ t d’1 |φ(t)|dt.

d ’2 0

Now that we know about integrability, we can apply Fd’2 to Iφ. Using integration by parts

and (d/dz)[z ν Jν (z)] = z ν Jν’1 (z) (see Proposition 5.4) leads to

∞

Fd’2 (Iφ)(r ) = r ’(d’4)/2 (Iφ)(t)t (d’2)/2 J(d’4)/2 (r t)dt

0

∞

∞

= r ’(d’2)/2 (Iφ)(t)t (d’2)/2 J(d’2)/2 (r t) + φ(t)t d/2 J(d’2)/2 (r t)dt

0

0

= Fd φ(r ).

The boundary terms vanish for the following reasons. Because of the integrability of

t ’ Iφ(t)t d’3 we have at least Iφ(t) = O(t ’d+2 ) for t ’ ∞. The asymptotic be-

√

havior of the Bessel functions gives Jν (t) = O(1/ t) (see Proposition 5.6). Hence,

(Iφ)(t)t (d’2)/2 J(d’2)/2 (r t) = O(t ’(d’1)/2 ) for t ’ ∞ and vanishes at in¬nity. For the lower

bound we use the asymptotic behavior of the Bessel functions Jν (t) = O(t ν ) for ν ≥ 0 and

t ’ 0 together with the boundedness of Iφ to derive (Iφ)(t)t (d’2)/2 J(d’2)/2 (r t) = O(t d’2 )

for t ’ 0, so that this function also vanishes at zero. This ¬nishes the proof of the ¬rst

part.

For the second part, de¬ne ψ := Dφ. Then ψ is well de¬ned, continuous, and satis¬es

t ’ ψ(t)t d+1 ∈ L 1 [0, ∞). This means in particular that Iψ = IDφ = φ. Finally, we can

apply the ¬rst part to ψ instead of φ and d + 1 instead of d to derive

Fd+2 (Dφ) = Fd+2 (ψ) = Fd (Iψ) = Fd (φ),

and this ¬nishes the proof.

This interaction between these operators allows us to express the higher-dimensional

Fourier transforms of radial functions by lower-dimensional ones and vice versa. Since

positive de¬nite integrable functions are characterized by a nonnegative and nonvanishing

Fourier transform we can draw the following conclusion.

Corollary 9.7 Suppose that φ is continuous. If on the one hand t ’ φ(t)t d’1 ∈ L 1 [0, ∞)

and d ≥ 3 then φ is positive de¬nite on Rd if and only if Iφ is positive de¬nite on Rd’2 . On

the other hand, if φ ∈ C 2 (R) is even and t ’ φ (t)t d ∈ L 1 [0, ∞) then φ is positive de¬nite

on Rd if and only if Dφ is positive de¬nite on Rd+2 .

9.3 Piecewise polynomial functions with local support 123

Proof This follows immediately from the preceding theorem and Bochner™s characteriza-

tion for radial and integrable functions given in Theorem 6.18.

The operators I and D that we have introduced vary the space dimension in steps of

width 2, which means that one deals with a sequence of either odd-dimensional spaces

or even-dimensional spaces. A generalization of the operators I and D to a whole family

of operators Iν with ν ∈ R where I1 = I and I’1 = D was made by Schaback and Wu

in [172]. This family allows us to walk through the space dimensions in the Fourier domain

not only in steps of width 2 but also in steps of width 1 and even, in a generalized way, in

steps of arbitrary width. Unfortunately, these operators no longer have a simple form and

thus are dif¬cult to apply.

9.3 Piecewise polynomial functions with local support

A local support of the basis function is only one step on the way to an ef¬cient numerical

approximation scheme. The next step is to ensure that the basis function is easily evaluated.

This is why from now on we will concentrate on functions of the form

0 ¤ r ¤ 1,

p(r ),

φ(r ) = (9.1)

r > 1,

0,

where p denotes a univariate polynomial. Of course, these functions are extended to the

whole real line, again by even extension. We can restrict ourselves to functions with support

in [0, 1] or [’1, 1], respectively. Other intervals can be obtained by scaling, because this

does not change a function from being positive de¬nite. The d-variate Fourier transform of

φ(·/δ), δ > 0, is δ d (Fd φ)(δ·), which is nonnegative if and only if the Fourier transform of

φ is nonnegative.

From Theorem 6.20 we already know a positive de¬nite function of the form (9.1): the

function

φ (r ) = (1 ’ r )+ (9.2)

is positive de¬nite on Rd provided that ≥ d/2 + 1.

These functions, when seen as even functions, are only continuous, even for large . Since

the basis function determines the smoothness of the approximant, it is necessary to have

smoother functions of the form (9.1) as well. Numerical considerations, however, ask for a

polynomial of the lowest possible degree. Hence it is quite natural to look for a function of

the form (9.1) with a polynomial of minimal degree, if its smoothness and space dimension

are prescribed. We will answer this question completely in the next section. But beforehand

we will give certain general results concerning functions of the form (9.1).

It is obvious that every even function φ of the form (9.1) possesses an even number

of continuous derivatives around zero and that this number is determined by the ¬rst odd

coef¬cient of the polynomial p that does not vanish. Furthermore φ is obviously in C ∞ at

(0, 1) and (1, ∞), so that the only critical point is 1.

124 Compactly supported functions

The proof of the following lemma, which describes the in¬‚uence of the operators I and

D on the smoothness, is straightforward if one takes the special form of φ into account.

One only has to integrate and differentiate polynomials.

Lemma 9.8 Suppose that φ is an even function of the form (9.1) and that it possesses 2k

continuous derivatives around 0 and continuous derivatives around 1. Then Iφ possesses

2k + 2 continuous derivatives around 0 and + 1 continuous derivatives around 1. If

k, ≥ 1 then Dφ possesses 2k ’ 2 continuous derivatives around 0 and ’ 1 continuous

derivatives around 1.

The results of Lemma 9.8 remain true for an arbitrary φ that is suf¬ciently smooth

outside 0 and 1 and allows the application of I and D. In such a situation, the only part that

needs a closer look is the smoothness at zero. For example, if φ is continuous at zero then