Let us provide an additional example and apply Theorem 6.11 to prove the positive

de¬niteness of another famous class of functions, namely inverse multiquadrics. Their

Fourier transforms involve the modi¬ed Bessel functions K ν .

Theorem 6.13 The function (x) = (c2 + x 2 )’β , x ∈ Rd , with c > 0 and β > d/2 is

2

positive de¬nite. It possesses the Fourier transform

β’d/2

ω

21’β 2

(ω) = K d/2’β (c ω 2 ). (6.6)

(β) c

Proof Since β > d/2 the function is in L 1 (Rd ). From the representation of the

-function for β > 0 we see that

∞

t β’1 e’t dt

(β) =

0

∞

β

u β’1 e’su du

=s

0

by substituting t = su with s > 0. If we set s = c2 + x 2

then we can conclude that

2

∞

1

u β’1 e’c u e’

2

2

(x) = x 2u du.

(β) 0

Inserting this into the Fourier transform and changing the order of integration, which can

easily be justi¬ed, leads to

(ω) = (2π )’d/2 (x)e’i x ω d x

T

Rd

∞

1

= (2π)’d/2 u β’1 e’c u e’ e’i x ω dud x

2

2 T

x 2u

(β) Rd 0

∞

1

= (2π)’d/2 u β’1 e’c e’ e’i x ω d xdu

2

2 T

u x 2u

(β) Rd

0

6.2 Bochner™s characterization 77

∞

1

u β’1 e’c u (2u)’d/2 e’ ω 2 /(4u)

2

= du

2

(β) 0

∞

1

u β’d/2’1 e’c u e’ ω 2 /(4u)

2

= du,

2

2d/2 (β) 0

where we have used Theorems 5.16 and 5.18. Moreover, we know from the proof of Lemma

5.14 that

∞

1

’ν

e(’r/2)(u/a+a/u) u ν’1 du

K ν (r ) = a

2 0

for every a > 0. If we now set r = c ω 2 , a = ω 2 /(2c), and ν = β ’ d/2 for ω = 0, we

derive

∞

d/2’β

ω2

1

e’uc e(’ ω 2 )/(4u) β’d/2’1

2

K β’d/2 (c ω 2 ) = u du

2

2 2c 0

d/2’β

ω

= 2β’1 (β)

2

(ω),

c

which leads to the stated Fourier transform for ω = 0 using K ’ν = K ν . We can use continu-

ity to see that (6.6) also holds for ω = 0. Since the modi¬ed Bessel function is nonnegative

and nonvanishing, is positive de¬nite.

Examples of inverse multiquadrics for β = 1/2 and 3/2 and c = 1 are provided on the

right-hand side of Figure 6.1.

We made the restriction β > d/2 to ensure that is integrable. This restriction makes

the function dependent in a certain way on the space dimension. Later we will see that this

restriction is arti¬cial and that any β > 0 leads to a positive de¬nite function on any Rd .

We want to close this section by considering a remarkable property of positive de¬nite

functions concerning their smoothness.

Theorem 6.14 Suppose that is a positive de¬nite function that belongs to C 2k in some

neighborhood of the origin. Then is in C 2k everywhere.

is positive de¬nite there exists a ¬nite nonnegative Borel measure μ on Rd

Proof Since

such that

(x) = (2π)’d/2 e’i x ω dμ(ω).

T

(6.7)

Rd

∞

This means that, for every test function γ ∈ C0 (Rd ),

(x)γ (x)d x = γ (ω)dμ(ω). (6.8)

Rd Rd

Again, we choose a regularization g (x) = d g( x) with a nonnegative function g ∈

∞

C0 (Rd ) with g L 1 (Rd ) = 1 and support {x ∈ Rd : x 2 ¤ 1}. Let denote the usual

d

= j=1 ‚ /‚x j , and

2

2 k

Laplace operator, i.e. the iterated Laplacian. By inserting

78 Positive de¬nite functions

(1 ’ )k g into (6.8) we ¬nd

g (ω)(1 + ω 2 )k dμ(ω) = (x)(1 ’ )k g (x)d x

2

Rd Rd