j,k=1

N

aβ

± j ±k (x j ’ xk )β

= (’1)m

β!

|β|=2m j,k=1

γ

±jxj ν

±k x k

N N

|ν|

= (’1) m

aβ (’1)

γ! ν!

γ +ν=β

|β|=2m j=1 k=1

= aγ +ν Aγ Aν

|γ |=m |ν|=m

≥ 0,

8.3 Examples of generalized Fourier transforms 109

γ

with Aγ = N ± j x j /γ !, which vanishes for |γ | < m. Thus we have by Proposition 8.4

j=1

and Lemma 8.11

2

N N

’i x T ω

± j ±k (x j ’ xk ) = ±je dμ(ω)

j

Rd \{0}

j,k=1 j=1

N

aβ

[’i(x j ’ xk )]β

+ ± j ±k

β!

|β|¤2m

j,k=1

≥ 0.

8.3 Examples of generalized Fourier transforms

In this section we will compute the generalized Fourier transforms of the most popular

basis functions. They can be used to show that the basis functions are conditionally positive

de¬nite. Even if the latter follows in most cases more easily from a characterization given in

the next section, the knowledge of the generalized Fourier transforms is of great importance

for error estimates and for estimates on the stability of the interpolation process to be derived

in the chapters that follow.

Our ¬rst example concerns the generalized Fourier transform of the multiquadrics. The

basic idea of the proof is to start with the classical Fourier transform of the inverse mul-

tiquadrics given in Theorem 6.13 and then to use analytic continuation. We will use the

notation t for the smallest integer greater than or equal to t ∈ R.

Theorem 8.15 The function (x) = (c2 + x 2 )β , x ∈ Rd , with c > 0 and β ∈ R \ N0

2

possesses the (generalized) Fourier transform

’β’d/2

ω

21+β 2

(ω) = K d/2+β (c ω 2 ), ω = 0, (8.7)

(’β) c

of order m = max(0, β ).

Proof De¬ne G = {» ∈ C : (») < m} and denote the right-hand side of (8.7) by •β (ω).

We are going to show by analytic continuation that

= •» (ω)γ (ω)dω, γ ∈ S2m

» (ω)γ (ω)dω (8.8)

Rd Rd

is valid for all » ∈ G where » (ω) = (c2 + ω 2 )» . First of all, note that (8.8) is valid for

2

» ∈ G with » < ’d/2 by Theorem 6.13 and in the case m > 0 also for » = 0, 1, . . . , m ’ 1

by Proposition 8.10 and the fact that 1/ (’») is zero in these cases. Analytic continuation

will lead us to our stated result when we can show that both sides of (8.8) exist and are

analytic functions in ». We will do this only for the right-hand side, since it is obvious for

110 Conditionally positive de¬nite functions

the left-hand side. Thus let us de¬ne

f (») = •» (ω)γ (ω)dω.

Rd

Suppose C is a closed curve in G. Since •» is an analytic function in » ∈ G it has the

representation

•z (ω)

1

•» (ω) = dz

z’»

2πi C

for » in the interior Int C of the curve C. Now suppose that we have already shown that the

integrand in the de¬nition of f (») can be bounded uniformly on C by an integrable function.

This ensures that f (») is well de¬ned in G and by Fubini™s theorem we can conclude

that

f (») = •» (ω)γ (ω)dω

Rd

•z (ω)

1

= dzγ (ω)dω

z’»

2πi Rd C

1 1

= •z (ω)γ (ω)dωdz

z ’ » Rd

2πi C

1 f (z)

= dz

z’»

2πi C

for » ∈ Int C, which means that f is analytic in G. Thus it remains to bound the integrand

uniformly. Let us ¬rst consider the asymptotic behavior in a neighborhood of the origin,

say for ω 2 < min{1/c, 1}. If we set b = (») we can use Lemma 5.14 to get, in the case

b = ’d/2,

2b+|b+d/2| (|b + d/2|) b+d/2’|b+d/2| ’b’d/2’|b+d/2|+2m

|•» (ω)γ (ω)| ¤ Cγ ω ,

c

| (’»)| 2

and, in the case b = ’d/2,

cω

21’d/2 1 2

|•» (ω)γ (ω)| ¤ Cγ ’ log ω 2.

2m

| (’»)| e 2

Since C is compact and 1/ is analytic this gives for all » ∈ C and ω < min{1/c, 1}

2

cω

’d+2 2

|•» (ω)γ (ω)| ¤ Cγ ,m,c,C 1 + ω ’ log ,

2

2

with = m ’ b > 0. For large arguments the integrand in the de¬nition of f (») can be

estimated by Lemma 5.13:

√

21+b 2π b+(d’1)/2 ’b’(d+1)/2 ’c ω 2 |b+d/2|2 /(2c ω 2 )

|•» (ω)γ (ω)| ¤ Cγ ω2 ,