Let us analyze the behavior of these four double sums as N ’ ∞. The ¬rst is a Riemann

102 Conditionally positive de¬nite functions

sum for K K γ (x)γ (y) (x ’ y)d xd y and hence converges to this integral. The second

and third double sums can each be bounded by

Q

1

|β (N ) |,

γ L ∞ ([’1,1]d )

L ∞ (K ) j

N j=1

and they tend to zero as N ’ ∞. The last double sum can be bounded by

2

Q

|β (N ) |

L ∞ ([’1,1]d ) j

j=1

and also tends to zero as N ’ ∞. This proves that the integral in (8.5) is indeed non-

negative.

We end this section with an interesting construction. A conditionally positive semi-

de¬nite function of order m can easily be used to construct a conditionally positive semi-

de¬nite function of order less than m.

Proposition 8.7 Suppose that is conditionally positive semi-de¬nite of order m > 0 and

that ¤ m is ¬xed. If y1 , . . . , y M ∈ Rd and β ∈ C M \ {0} satisfy M β j p(y j ) = 0 for all

j=1

p ∈ π ’1 (Rd ) then the function

M

β j βk (x ’ y j + yk )

(x) :=

j,k=1

is conditionally positive semi-de¬nite of order m ’ .

Proof We will use the monomials as a basis for the polynomial space. Suppose

ν

N

x1 , . . . , x N ∈ Rd and ± ∈ C N are given so that j=1 ± j x j = 0 for all ν ∈ N0 with

d

|ν| < m ’ . Then

N N M

± ±n (x ’ xn ) = ± β j ±n βk ((x ’ y j ) ’ (xn ’ yk ))

,n=1 ,n=1 j,k=1

= C I C J (z I ’ z J ),

I J

where each of the last sums runs over M N terms, C I = ± β j , and z I = x ’ y j . The last

expression is nonnegative by the assumptions imposed on , if we can show that the new

centers and new coef¬cients satisfy the side conditions for polynomials of degree less than

m. But this is true because

N M

C I zν ± β j (x ’ y j )ν

=

I

=1 j=1

I

ν N M

ν’μ μ

(’1)|μ|

= ±x βj yj

μ

μ¤ν =1 j=1

= 0.

8.2 An analogue of Bochner™s characterization 103

The last equality holds for all |ν| < m because either |ν ’ μ| < m ’ or |μ| < is

satis¬ed.

8.2 An analogue of Bochner™s characterization

In the case of positive de¬nite functions we found the integral characterization by Bochner

to be very helpful. For all relevant basis functions, the version given in Theorem 6.11 for

integrable functions was suf¬cient.

For a conditionally positive de¬nite function also there exists a characterization compa-

rable to Bochner™s and we will state it at the end of this section. But we want to start with

a version that is, rather, an analogue of the result in Theorem 6.11. Of course, for a con-

ditionally positive de¬nite function we cannot hope for integrability. But the crucial point

in the positive de¬nite case was actually not integrability but the existence of a classical

Fourier transform. If we want to apply this idea here, we have to modify the notion of the

Fourier transform for our purposes. To this end, a special subspace of the Schwartz space

S will be of importance.

De¬nition 8.8 For m ∈ N0 the set of all functions γ ∈ S that satisfy γ (ω) = O( ω m

2) for

ω 2 ’ 0 will be denoted by Sm .

In what follows, we restrict ourselves to slowly increasing basis functions. The reader

should remember that a function is called slowly increasing if it grows at most like any

particular ¬xed polynomial. To discuss only such functions is actually not a restriction,

because one can show that every conditionally positive de¬nite function of order m grows

at most like a polynomial of degree 2m (see Madych and Nelson [113]).

De¬nition 8.9 Suppose that : Rd ’ C is continuous and slowly increasing. A measur-

able function ∈ L loc (Rd \ {0}) is called the generalized Fourier transform of if there

2

exists an integer m ∈ N0 such that

(x)γ (x)d x = (ω)γ (ω)dω

Rd Rd

is satis¬ed for all γ ∈ S2m . The integer m is called the order of .

Note that the order m of a generalized Fourier transform corresponds to the space S2m

and not Sm . Furthermore, if is a generalized Fourier transform of order m then it has also

order ≥ m. Hence, in general we will refer to the smallest possible m when speaking of

the order.

Several remarks are necessary. If the generalized Fourier transform exists in this way, it

is uniquely determined up to Lebesgue-zero sets. If ∈ L 1 (Rd ) then its classical Fourier

transform and its generalized Fourier transform coincide. The order is zero. The same is true

for ∈ L 2 (Rd ). The generalized Fourier transform and the distributional Fourier transform

coincide on the set S2m .

In this chapter, we are concerned only with generalized Fourier transforms that are

continuous on Rd \ {0} and have an algebraic singularity at the origin. The order of the

104 Conditionally positive de¬nite functions

singularity determines the minimal order m of the generalized Fourier transform. Later, we

will need the more general form.

The next result not only gives an example of a generalized Fourier transform, it also

shows in which way the function is determined by its generalized Fourier transform.

Proposition 8.10 Suppose = p is a polynomial of degree less than m. Then for every

test function γ ∈ Sm we have

(x)γ (x)d x = 0. (8.6)

Rd

Hence the generalized Fourier transform of p is the zero function and has order m/2.

is a continuous function satisfying (8.6) for all γ ∈ Sm then

Conversely, if is a

polynomial of degree less than m.

cβ x β .

(x) =

Proof For the ¬rst part let us assume that has the representation |β|<m

Then

cβ i ’|β| (i x)β γ (x)d x

(x)γ (x)d x =

Rd Rd

|β|<m

cβ i ’|β| D β γ (x)d x

=

Rd

|β|<m

cβ i ’|β| D β γ (0)

= (2π )d/2

|β|<m

= 0,