After investigating the native spaces for the compactly supported functions of mini-

mal degree we now turn to another famous class of radial basis functions, the thin-plate

splines. To be more precise we want to characterize the native spaces of the functions

d, := φd, ( · 2 ) with > d/2 and

§

(d/2 ’ ) 2 ’d

⎪

⎪ 2 d/2 ,

⎨ 2 π ( ’ 1)! r for d odd,

φd, (r ) := (10.11)

(’1) +(d’2)/2

⎪

⎪ 2 ’d

© r log r for d even.

22 ’1 π d/2 ( ’ 1)!( ’ d/2)!

10.5 Special cases of native spaces 161

From Theorems 8.16 and 8.17 we know that has a generalized Fourier transform

d,

’2

(ω) = (2π)’d/2 ω

d, 2

of order m = ’ d/2 + 1, so that d, is conditionally positive de¬nite of order m.

In contrast with our earlier convention but in accordance with Proposition 8.2 we will

consider d, as a conditionally positive de¬nite function of order and its generalized

Fourier transform as of order for the rest of this section.

The reason for choosing the constant factor in this way is given by the simple structure of

the Fourier transform, which leads to the fact that d, is a fundamental solution of the iter-

ated Laplacian. Remember that the Laplacian operator is de¬ned to be := d ‚ 2 /‚x2 j=1 j

’1

and the iterated Laplacian to be := .

Theorem 10.36 Let d, ∈ N with > d/2. If := φd, ( · 2 ) with the univariate func-

d,

tion φd, from (10.11) then

g(x ’ ω)dω = g(x)

(’1) (ω)

d,

Rd

for all g ∈ S and x ∈ Rd .

Proof De¬ne γ by its Fourier transform γ (ω) := g(x ’ ω). Then γ is given by

γ (ω) = (2π )’d/2 g(x ’ ·)ei· ω d·

T

Rd

ixT ω §

=e ( g) (ω)

ω

T

= (’1) ei x ω 2

g(ω),

2

showing that γ ∈ S2 for every g ∈ S and x ∈ Rd . Hence, we can invoke the theory on

generalized Fourier transforms to derive

g(x ’ ω)dω = (2π)’d/2 ω

T

g(ω)dω = g(x),

ei x

(’1) (ω)

d,

Rd Rd

using the special form of the Fourier transform of mentioned earlier.

d,

Next we introduce Beppo Levi spaces. To this end we have to de¬ne the generalized

derivative of a continuous function.

De¬nition 10.37 Let f ∈ L loc (Rd ) and ± ∈ Nd be given. A function f ± ∈ L loc (Rd ) is the

1 1

generalized (or weak) derivative of f of order ± if

f (x)D ± γ (x)d x = (’1)|±| f ± (x)γ (x)d x (10.12)

Rd Rd

∞

is satis¬ed for all γ ∈ C0 (Rd ). We will use the notation D ± f := f ± again.

For > d/2, the linear space

BL (Rd ) := { f ∈ C(Rd ) : D ± f ∈ L 2 (Rd ) for all |±| = }

162 Native spaces

equipped with the inner product

!±

(D f, D ± g) L 2 (Rd )

( f, g)BL (Rd ) :=

±!

|±|=

is called the Beppo Levi space on Rd of order .

Beppo Levi spaces can be introduced in a much more general way. The most general

∞

version starts with D = C0 ( ), ⊆ Rd , and its dual D , the set of distributions. The

advanced reader will know what type of continuity is meant in the de¬nition of the dual

space. Next, one chooses a separable complete space E of functions de¬ned on and

de¬nes the Beppo Levi space to be

BL (E) := { f ∈ D : D ± f ∈ E for all |±| = }.

In our speci¬c situation it is possible to show that the two de¬nitions coincide, i.e. BL (Rd ) =

BL (L 2 (Rd )). Details may be found in the papers by Deny and Lions [45], Duchon [47],

and Light and Wayne [108] and the other sources on Beppo Levi spaces cited in the

references.

The choice of weights !/±! in the de¬nition is motivated by expressing x 2 as x 2 =

2 2

!x 2± /±!. This also means that we can express the iterated Laplacian by =

|±|=

|±|= !D /±!. Both will be important later on.

2±

The rest of this section is devoted to showing that the native space of d, is the Beppo

Levi space BL (Rd ). We start by showing that the null space of the semi-inner product is

the space of polynomials of degree less than . Clearly, π ’1 (Rd ) is in the null space of

(·, ·)BL (Rd ) . It remains to prove that they are actually the same.

Lemma 10.38 Suppose that f ∈ BL (Rd ), > d/2, satis¬es D ± f = 0 for all |±| = .

Then f is a polynomial of degree less than .

∞

Proof We use approximation by convolution. Let g ∈ C0 (Rd ) be nonnegative and even,

having integral one. Set gn := n d g(n ·) as usual. Then we know from Theorem 5.22 that

f — gn ∈ C ∞ (Rd ) and D ± ( f — gn ) = f — (D ± gn ). An application of the de¬nition of the

generalized derivative gives immediately D ± ( f — gn ) = (D ± f ) — gn = 0 for |±| = . Hence

for all n ∈ N and all |±| = the C ∞ -functions D ± ( f — gn ) are zero, implying that f — gn ∈

π ’1 (Rd ) for all n. Moreover, f — gn (x) tends to f (x) as n ’ ∞ for all x ∈ Rd . But if we ¬x

x ∈ Rd the latter convergence means the convergence of the coef¬cients of the polynomials

f — gn ∈ π ’1 (Rd ). Thus f is also a polynomial of degree less than .

The next step is to show that the native space of the thin-plate splines d, is contained in

the Beppo Levi space B L (Rd ), and that on N d, (Rd ) both semi-inner products are equal.

To this end we will use the Fourier transform representation of N d, (Rd ) given in Theorem

10.21.

Proposition 10.39 For > d/2 let d, = φd, ( · 2 ) be the thin-plate spline de¬ned in

(10.11). If d, is considered to be a conditionally positive de¬nite function of order

10.5 Special cases of native spaces 163

then the associated native space is contained in the Beppo Levi space of order , i.e.

N d, (Rd ) ≡ N d, ,π ’1 (Rd ) (Rd ) ⊆ BL (Rd ), and the semi-inner products are the same on

this subspace.