•n ’ •m = •n + •m ’ 2(•n , •m ) ’ c + c ’ 2c = 0

2 2 2

168 Native spaces

for m, n ’ ∞. Thus there exists a • ∈ F ( ) with • ’ •n ’ 0 for n ’ ∞. For this

element we make the computation

R•(y) = (•, G(·, y))

= lim (•n , G(·, y))

n’∞

= lim (•n (y) ’ P •n (y))

n’∞

± ±

= ’ D2 G(u, x) |u=x ,

u

D2 G(y, x) P

±

showing that D2 G(·, x) indeed belongs to N ( ).

With this lemma at hand it is easy to prove the smoothness of the functions belonging to

the native space of a smooth basis function.

Theorem 10.45 Suppose that ⊆ Rd is open and that ∈ C 2k ( — ) is a conditionally

positive de¬nite kernel with respect to P ⊆ C k ( ); then N ( ) ⊆ C k ( ) and, for every

f ∈ N ( ), every ± ∈ Nd with |±| ¤ k, and every x ∈ , we have the representation

0

D ± f (x) = D ± ±

f (x) + f, D2 G(·, x) . (10.15)

P N()

Proof We will show (10.15) by induction on |±|. This will obviously prove the existence

and continuity of the derivatives. For |±| = 0, formula (10.15) obviously coincides with

the representation in Theorem 10.17. For |±| > 0 we can assume that ±1 > 0. Hence, with

β = (±1 ’ 1, ±2 , . . . , ±d )T we ¬nd that

1

D ± f (x) = lim D β f (x + he1 ) ’ D β f (x)

h’0 h

1

D β ( P f )(x + he1 ) ’ D β ( P f )(x)

= lim

h’0 h

1 β β

+ lim f, D2 G(·, x + he1 ) ’ D2 G(·, x)

h

h’0 N()

± ±

=D ( f )(x) + f, ,

D2 G(·, x) N ( )

P

using the fact that the derivatives of G(·, x) exist by Lemma 10.44. As usual e1 denotes the

¬rst unit vector in Rd .

In the situation where is a positive de¬nite function (i.e. a translation-invariant kernel)

which is in L 1 (Rd ) and which has a Fourier transform that decays like (1 + · 2 )’s we

2

know by Corollary 10.13 that the native space is actually the Sobolev space H (Rd ). If

s

s > k + d/2 then the Fourier inversion formula guarantees that ∈ C 2k (Rd ) and Theorem

10.45 shows that H s (Rd ) ⊆ C k (Rd ), which is Sobolev™s embedding theorem.

10.7 Restriction and extension

In this section we want to investigate how the native space N ( ) depends on the region

. To do this we have to be more careful about the notation, even if sometimes it seems

excessive.

10.7 Restriction and extension 169

Let us assume that we are dealing with two regions that satisfy 1 ⊆ 2 ⊆ Rd . We are

now interested in the questions whether the functions from N ( 1 ) have an extension to 2

and whether the restrictions of the functions from N ( 2 ) to 1 lie in N ( 1 ). Of course,

both should be true and we shall prove the results in this section.

The crucial point in everything we do here is that we assume that the set is already

contained in 1 , that P ⊆ C( 2 ), and that ∈ C( 2 — 2 ) is a conditionally positive

de¬nite kernel with respect to P on 2 . In the case of a positive de¬nite kernel we need

only the last assumption, that is positive de¬nite on the larger set.

Theorem 10.46 Each function f ∈ N ( 1 ) has a natural extension to a function E f ∈

N ( 2 ). Furthermore, |E f |N ( 2 ) = | f |N ( 1 ) .

Proof Since 1 ⊆ 2 we have a natural extension : F ( 1 ) ’ F ( 2 ), simply by eval-

uation of a function f ∈ F ( 1 ) at points from 2 also. Since the norm f , 1 depends

only on the centers and coef¬cients of f , we have obviously f , 2 = f , 1 . Hence

is an isometric embedding that has an continuous extension : F ( 1 ) ’ F ( 2 ).

This allows us to construct the extension operator E : N ( 1 ) ’ N ( 2 ) in the follow-

ing way. Every f ∈ N ( 1 ) has the representation f (x) = P f (x) + R 1 ( f˜)(x), with

f˜ ∈ F ( 1 ). For this f and x ∈ 2 we de¬ne

E f (x) = f (x) + R 2 ( f˜)(x).

P

The function P f has an obvious extension to 2. This is why we did not use different

notation. Moreover, for x ∈ 1 we have

R 2 ( f˜)(x) = ( f˜, G 2 (·, x)) , 2 = ( f˜, G 1 (·, x)) , 2

= ( f˜, G 1 (·, x)) , 1 ,

showing that E f (x) = f (x) for f ∈ N ( and x ∈ Finally, for two functions f, g ∈

1) 1.

N ( 1 ) the identities

= ( f˜, g) = ( f˜, g) = ( f, g)N

˜ ˜

(E f, Eg)N , ,

( 2) ( 1)

2 1

show that E is isometric.

Now let us turn to the restriction of a function f ∈ N ( 2 ) to 1 . By Theorem 10.26,

f | 1 ∈ N ( 1 ) if there exists a constant c f such that |»( f | 1 )| ¤ c f » , 1 for all

» ∈ L P ( 1 ). Since obviously L P ( 1 ) ⊆ L P ( 2 ) it is true that there exists a constant c f

with

|»( f | ¤ cf » = cf » ,

1 )| , ,

2 1

giving f | ∈N . Finally,

1 ( 1)

|»( f )| |»( f )|

f| = sup ¤ sup =f

1N( N(

1) 2)

» ,2 » ,2

»∈L P ( 1 ) »∈L P ( 2 )

»=0 »=0

¬nishes the proof of our next theorem.

170 Native spaces

Theorem 10.47 The restriction f | 1 of any function f ∈ N ( 2 ) is contained in N ( 1)

with a semi-norm that is less than or equal to the semi-norm of f .

The concept of restriction and extension has an interesting implication in the case of

Sobolev spaces. We already know that the native space over the entire Rd of a basis function

with algebraically decaying Fourier transform is a Sobolev space. Now we are able to show

that this is also true for “nice” regions ⊆ Rd . To this end, let us recall that the Sobolev

space H k ( ), k ∈ N, for a measurable set can be introduced using weak derivatives. It

consists of all functions f ∈ L 2 ( ) having weak derivatives in L 2 ( ) of order |±| ¤ k. The

norm on H k ( ) is then given by u 2 k ( ) = |±|¤k D ± u 2 2 ( ) . In the case = Rd it is

L

H

known that this norm is equivalent to the norm previously de¬ned by Fourier transformation.