t∈Tρ |±|=m t∈Tρ

q/ p

p

¤ (#Tr )q(1/q’1/ p)+ |u|Wqm (Dt ) ,

t∈Tρ

where the last bound follows from standard inequalities relating the p-norm and the q-norm

on ¬nite-dimensional spaces. Proposition 11.30 together with dDt ¤ 2R = 2Q(k, θ )’1 h

11.6 Sobolev bounds for functions with scattered zeros 201

gives the bound

1/ p 1/ p

p p

|u|Wqm (Dt ) ¤ Ch |u|W k+s (D ) .

k+s’m+d(1/q’1/ p)

p t

t∈Tρ t∈Tr

Here, it has been essential that all involved constants depend only on the cone condition,

which is the same for all Dt . Now using Lemma 11.31 again yields

|D ± u(x) ’ D ± u(y)| p

p

|u|W k+s (D ) ¤ χDt (x) d yd x

d+sp

x’y

p t

Dt

|±|=k

t∈Tρ t∈Tρ 2

|D ± u(x) ’ D ± u(y)| p

¤ M1 d yd x

d+sp

x’y

|±|=k 2

p

¤ M1 |u|W k+s ( ) .

p

A ¬nal application of Lemma 11.31 together with

sin θ

ρ= h

2Q(k, θ )(1 + sin θ)

shows that #Tρ ¤ Ch ’d . Putting all these things together and taking

1 1 1 1 1 1

’ ’d ’ = ’d ’

d

q p q p pq

+ +

into account establishes the desired result.

We end this section by applying the last theorem to radial basis functions of compact

support and to thin-plate splines.

Let d,k = φd,k ( · 2 ) be the compactly supported basis functions from De¬nition 9.11.

If ⊆ Rd has a Lipschitz boundary, we know by Theorem 10.35 and Corollary 10.48,

which can be extended to the case of fractional Sobolev spaces (see the discussion in [31]),

that the associated native space is norm-equivalent to the Sobolev space H k+(d+1)/2 ( ) =

k+(d+1)/2

( ). We treat this case in the more general situation where ∈ L 1 (Rd ) has a

W2

Fourier transform that satis¬es

c1 (1 + ω 2 )’„ ¤ (ω) ¤ c2 (1 + ω 2 )’„ , ω ∈ Rd , (11.20)

2 2

with „ > d/2.

Corollary 11.33 Suppose that ⊆ Rd is bounded, has a Lipschitz boundary, and satis¬es

an interior cone condition with radius r and angle θ . Let X ⊆ be a given discrete set

of centers and s f,X be the interpolant. Suppose that satis¬es (11.20) with „ = k + s,

where k is a positive integer and 0 < s ¤ 1. If m ∈ N0 satis¬es k > m + d/2 then the error

„

between f ∈ W2 ( ) and its interpolant s f,X can be bounded by

„ ’m’d(1/2’1/q)+

| f ’ s f,X |Wqm ( ¤ Ch X, f „

) W2 ( )

for all suf¬ciently dense sets X .

202 Error estimates for radial basis function interpolation

Next, we want to apply these results to the thin-plate splines d, = φd, ( · 2 ) from

(10.11). We know by Theorem 10.43 that the global native space N d, (Rd ) is the Beppo

Levi space BL (Rd ). To apply Theorem 11.32, we need an extension operator from BL ( )

to BL (Rd ).

Lemma 11.34 Suppose that ⊆ Rd is open and bounded and satis¬es an interior cone

condition. For every f ∈ BL (Rd ), > d/2, there exists a unique function f ∈ BL (Rd )

with f | = f and

| f |BL (Rd ) = min{|g|BL (Rd ) : g ∈ BL (Rd ) = f | }.

and g|

Proof Fix a π (Rd )-unisolvent set = {ξ1 , . . . , ξ Q } ⊆ and introduce the inner product

Q

( f, g)Rd := ( f, g)BL (Rd ) + f (ξ j )g(ξ j ).

j=1

With this inner product, BL (Rd ) becomes a reproducing-kernel Hilbert space (see Theorem

10.20). Moreover, since all relevant functions coincide with f when restricted to , the

minimization problem is equivalent to minimizing the norm · Rd on

V f = {g ∈ BL (Rd ) : g| = f | }.

But this set is obviously nonempty since it contains f , it is convex, and it is closed. The

last follows from the reproducing-kernel property. If {gn } ⊆ V f converges to g ∈ BL (Rd )

then the reproducing kernel gives also pointwise convergence, i.e. gn (x) ’ g(x), x ∈ Rd .

This means g| = f .

Moreover, the minimization problem amounts to nothing other than ¬nding the best

approximation from V f to 0. Because of the properties of V f just stated, this is uniquely

solvable.

Lemma 11.35 Let > d/2 and ⊆ Rd be open and bounded, satisfying an interior cone

condition. For every f ∈ H ( ) there exists a unique | · |BL (Rd ) minimal extension f ∈

BL (Rd ). Moreover, this extension is continuous, i.e. there exists a constant K > 0 such

that

| f |BL (Rd ) ¤ K | f |BL ( ) .

Proof Since ⊆ Rd satis¬es the cone condition there is a continuous extension operator

from H ( ) to H (Rd ), meaning that there exists a constant C > 0 such that we can ¬nd

for every f ∈ H ( ) a function f ∈ H (Rd ) with f | = f and f H (Rd ) ¤ C f H ( ) .

Since obviously H (Rd ) ⊆ BL (Rd ), Lemma 11.34 gives us a function f ∈ BL (Rd )

which coincides on with f and which has a minimal Beppo Levi semi-norm amongst all

such functions. The uniqueness follows from Lemma 11.34 and the fact that all possible

extensions f of f coincide with f on .

11.6 Sobolev bounds for functions with scattered zeros 203

By the proof of that lemma we even know that f has minimal · Rd -norm with

f 2 d := | f |2 (Rd ) + | f (ξ j )|2 . Hence, using the Sobolev embedding theorem, we have

R BL

¤f ¤C f ¤C f

f Rd Rd H (Rd ) H( )

with some generic constant C > 0. If we can show that · de¬ned by f 2 =

| f |BL ( ) + | f (ξ j )|2 is equivalent to · H ( ) on H ( ) then we can immediately derive

f Rd ¤ C f and hence, since ⊆ , also | f |BL (Rd ) ¤ C| f |BL ( ) . To show norm

equivalence we use standard arguments from the theory of Sobolev spaces. First of all, ·

can obviously be bounded by a constant times · H ( ) , again by the Sobolev embedding

theorem. Unfortunately this is not the inequality we need. To prove the other inequality let

us assume that it is wrong. Then we can ¬nd a sequence {•n }n∈N with •n H ( ) = 1 and

Q

!

D ± •n

1 = •n >n + |•n (ξ j )|2 .

2 2

(11.21)

±! L2( )

H( )

|±|= j=1