= (ω)γ (ω)dω

Rd

2

N

iω T x j

= ±je g (ω) (ω)dω

Rd j=1

≥ 0.

Thus we have by Theorem 5.20,

N

± j ±k (x j ’ xk ) = lim f (x)g (x)d x ≥ 0.

’∞ Rd

j,k=1

2

N T

j=1 ± j e g (ω) (ω) is nondecreasing in ∈ N, the Beppo“

iω x j

Moreover, since

Levi convergence theorem guarantees the integrability of the limit function

2

(2π)’d/2 N T

± j eiω xj

(ω) and also the identity

j=1

2

N N

± j ±k (x j ’ xk ) = (2π )’d/2

T

± j eiω xj

(ω)dω.

Rd

j,k=1 j=1

8.2 An analogue of Bochner™s characterization 107

The same arguments as in the proof of Theorem 6.11 show that the quadratic form cannot

vanish if is nonvanishing.

Now suppose that is conditionally positive de¬nite of order m. Because of Proposition

∞

8.6, the function satis¬es (8.3) for all γ ∈ C0 (Rd ) with (8.4).

∞

Next choose a nonnegative function k ∈ C0 (Rd ) having support B(0, 1) := {x ∈ Rd :

x 2 ¤ 1} with k 2 2 (Rd ) = (2π )’d/2 . If we set k (x) := d/2 k( x), x ∈ Rd , and

L

γ (x) := k (· ’ y)§ (x) = e’i x y k (x)

T

for a ¬xed y = 0, we ¬nd by application of Theorem 5.16, for every multi-index

± ∈ Nd ,

0

γ (x)x ± d x = x ± k (x)e’i x y d x

T

Rd Rd

= (2π)d/2 i ’|±| (D ± k )(’y)

= 0,

provided that > 1/ y 2 . Thus on the one hand γ satis¬es (8.4) for these -values and

can be inserted into (8.3), which gives

(x)γ — γ (x)d x ≥ 0

Rd

with γ (x) := γ (’x). On the other hand we have

(γ — γ )∨ (0) = (2π)d/2 |k (’y)|2 = 0

if > 1/ y 2 . Thus we can conclude that (γ — γ )∨ lies in S2m and can be inserted into the

de¬nition of the generalized Fourier transform. Using

(γ — γ )∨ (x) = (2π)d/2 |(γ )∨ (x)|2 = (2π)d/2 |k (x ’ y)|2

leads to

0¤ (x ’ z)γ (x)γ (z)d xdz

Rd Rd

= (x)(γ — γ )(x)d x

Rd

(ω)(γ — γ )∨ (ω)dω

=

Rd

= (2π)d/2 (ω)|k (ω ’ y)|2 dω,

Rd

which converges for ’ ∞ to (y).

For reasons that will become clear later, we will restate the representation for the quadratic

form we derived in the last proof.

108 Conditionally positive de¬nite functions

Corollary 8.13 Suppose that : Rd ’ C is continuous and slowly increasing. Suppose

further that possesses a nonnegative, nonvanishing, generalized Fourier transform of

order m that is continuous on Rd \ {0}. Then for all pairwise distinct x1 , . . . , x N ∈ Rd and

all ± ∈ C N with (8.1) for all p ∈ πm’1 (Rd ), we have

2

N N

’d/2 iω T x j

± j ±k (x j ’ xk ) = (2π ) ±je (ω)dω.

Rd

j,k=1 j=1

Theorem 8.12 will be suf¬cient for all our goals. Nonetheless, one might be interested

in ¬nding a complete characterization of all conditionally positive semi-de¬nite functions

of a given order m. For completeness, we will state the result here and prove the suf-

¬cient part. For the proof of the necessary part, we refer the interested reader to Sun™s

paper [182].

Theorem 8.14 Let ∈ C(Rd ). In order for to be conditionally positive semi-de¬nite of

order m it is necessary and suf¬cient that has the following integral representation:

(’i x)β (’i x)β

e’i x ω

T

(x) = ’ κ(ω) dμ(ω) + .

aβ

β! β!

Rd \{0} |β|<2m |β|¤2m

Here μ is a positive Borel measure on Rd \ {0} satisfying

ω <∞ dμ(ω) < ∞.

2m

2 dμ(ω) and

0< ω 2 ¤1 ω 2 ≥1

The function κ is an analytic function in S such that κ(ω) ’ 1 has a zero of order 2m + 1 at

the origin. The numbers aβ , |β| = 2m, satisfy |β|=m,|γ |=m ±β ±γ aβ+γ ≥ 0 for all ±β ∈ C.

Proof As stated previously we want to prove only the suf¬cient part of this theorem.

Suppose x1 , . . . , x N ∈ Rd and ± ∈ C N with (8.1) are given. From Proposition 8.4 we know

that

N

aβ

[’i(x j ’ xk )]β

± j ±k

β!