| (’»)|

8.3 Examples of generalized Fourier transforms 111

√

Fig. 8.1 The multiquadric φ(r ) = 1 + r 2 (on the left) and the thin-plate spline φ(r ) = |r |2 log(|r |)

(on the right).

using the fact that γ ∈ S is certainly bounded. Since C is compact, this can be bounded

independently of » ∈ C by

|•» (ω)γ (ω)| ¤ Cγ ,C,m,c e’c ω

.

2

This completes the proof.

√

The left-hand half of Figure 8.1 shows the function φ(r ) = 1 + r 2 , for which the name

multiquadric has been coined.

β

(x) = x x ∈ Rd , with β > 0, β ∈ 2N, has the general-

Theorem 8.16 The function 2,

ized Fourier transform

2β+d/2 ((d + β)/2) ’β’d

(ω) = ω , ω = 0,

2

(’β/2)

of order m = β/2 .

Proof Let us start with the function c (x) = (c2 + x 2 )β/2 , c > 0. This function pos-

2

sesses a generalized Fourier transform of order m = β/2 given by

21+β/2 ’β’d

c (ω) = •c (ω) = ω ω 2 )(β+d)/2 K (β+d)/2 (c ω 2 ),

(c

2

(’β/2)

owing to Theorem 8.15. Here we use the subscript c instead of β, since β is ¬xed and we

want to let c go to zero. Moreover, we can conclude from the proof of Theorem 8.15 that

for γ ∈ S2m the product can be bounded by

2β+d/2 ((β + d)/2) 2m’β’d

|•c (ω)γ (ω)| ¤ Cγ ω

| (’β/2)| 2

for ω ’ 0 and by

2

2β+d/2 ((β + d)/2) ’β’d

|•c (ω)γ (ω)| ¤ Cγ ω

| (’β/2)| 2

for ω 2 ’ ∞, independently of c > 0. Since | c (ω)γ (ω)| can also be bounded indepen-

dently of c by an integrable function, we can use the convergence theorem of Lebesgue

112 Conditionally positive de¬nite functions

twice to derive

β

2 γ (x)d x = lim

x c (x)γ (x)d x

c’0 Rd

Rd

= lim •c (ω)γ (ω)d x

c’0 Rd

γ (ω)

21+β/2

= lim (c ω 2 )(β+d)/2 K (β+d)/2 (c ω 2 )dω

β+d c’0

ω

(’β/2) Rd 2

2β+d/2 ((d + β)/2) ’β’d

= ω γ (ω)dω

2

(’β/2) Rd

for γ ∈ S2m . The last equality follows from

∞

ν ν’1

e’t e’r /(4t) ν’1

dt = 2ν’1 (ν);

2

lim r K ν (r ) = lim 2 t

r ’0 r ’0 0

see also the proof of Lemma 5.14.

Our ¬nal example deals with the thin-plate or surface splines. The right-hand half of

Figure 8.1 shows the most popular representative of this class.

(x) = x log x 2 , x ∈ Rd , k ∈ N, possesses the gener-

2k

Theorem 8.17 The function 2

alized Fourier transform

’d’2k

(ω) = (’1)k+1 22k’1+d/2 (k + d/2)k! ω 2

of order m = k + 1.

Proof For r > 0 ¬xed and β ∈ (2k, 2k + 1) we expand the function β ’ r β in a Taylor

series, obtaining

β

r β = r 2k + (β ’ 2k)r 2k log r + (β ’ t)r t log2 (r ) dt. (8.9)

2k

β

From Theorem 8.16 we know the generalized Fourier transform of the function x ’ x 2

of order m = β/2 = k + 1. From Proposition 8.10 we see that the generalized Fourier

transform of order m of the function x ’ x 2k equals zero. Thus we can conclude from

2

(8.9) for any test function γ ∈ S2m that

γ (x) x 2k

log x dx

2

2

Rd

1 β

= γ (x) ’x 2k

x dx

β ’ 2k 2

2

Rd

β

1

’ (β ’ t)γ (x) x t

log2 x dtd x

2

β ’ 2k 2

Rd 2k

2β+d/2 ((d + β)/2) ’β’d

= ω γ (ω)dω + O(β ’ 2k),

(β ’ 2k) (’β/2) 2

Rd

8.4 Radial conditionally positive de¬nite functions 113

for β ’ 2k. Furthermore, we know from Proposition 5.2 that

sin(πβ/2) (1 + β/2)

1

=’ .

(’β/2)(β ’ 2k) π(β ’ 2k)

Because

π π

sin(πβ/2)

= lim cos(πβ/2) = (’1)k

lim

β ’ 2k