Lemma 11.28 For 1 < p < ∞ and k > d/ p or p = 1 and k ≥ d, if u ∈ W p (D) and P ∈

k

πk (Rd ) then

k’d/ p

u ’ Q k+1 u ¤ C(1 + γ )d dD |u ’ P|W p (D) , (11.16)

k

L ∞ (D)

where C depends only on k, d, p.

Proof Since P ∈ πk (R) we have Q k+1 P = P. Hence, if we set v := u ’ P then we get

u ’ Q k+1 u = v ’ Q k+1 v

1

D ± v(y)(· ’ y)± φρ (y)dy.

= v ’ Qk v ’

±! Bρ

|±|=k

The term v ’ Q k v can be bounded using Proposition 11.27, while the rest enjoys the upper

bound

CdD ρ ’d max |D ± v(y)|dy ¤ CdD ρ ’d vol(Bρ )1’1/ p |v|W p (D)

k k

k

|±|=k Bρ

k’d/ p

¤ C(dD /ρ)d/ p dD |v|W p (D) .

k

Finally, dD /ρ ¤ 2γ and γ d/ p ¤ (1 + γ )d give the desired result.

11.6 Sobolev bounds for functions with scattered zeros 197

Now we are able to derive our result for fractional-order Sobolev spaces. Such a result could

also be proved by using interpolation theory on the operator Q k+1 , but here we will prove

it directly.

Proposition 11.29 Let 0 < s ¤ 1 and m ∈ N. Let 1 < p < ∞ and k > m + d/ p, or p = 1

and k ≥ m + d. For u ∈ W p (D) we have

k+s

k+s’m’d/ p

u ’ Q k+1 u ¤ C(1 + γ )d(1+1/ p) dD |u|W p (D)

m k+s

W∞ (D)

with a constant C > 0 depending only on k, d, p.

Proof We start with the situation m = 0. The case s = 1 follows immediately from Propo-

sition 11.27. Hence, we might assume that 0 < s < 1. In Lemma 11.28 let P = Q k+1 u ∈

πk (Rd ). The identity

D ± Q k+1 u = Q k+1’|±| D ± u, (11.17)

which is easily established, holds for |±| ¤ k. In particular, if we take |±| = k then we have

D ± Q k+1 u = Q 1 D ± u = φρ (y)D ± u(y)dy,

Bρ

which is of course a constant. Since Bρ φρ (y)dy = 1, we can use standard arguments,

x ’ y 2 ¤ dD , bounds on φρ , and H¨ lder™s inequality, to obtain the estimate

o

|D ± (u ’ Q k+1 u)(x)| ¤ φρ (y) |D ± u(x) ’ D ± u(y)| dy

Bρ

±

u(x) ’ D ± u(y)|

s+d/ p |D

¤ φρ (y) x ’ y dy

2 s+d/ p

x’y 2

Bρ

|D u(x) ’ D ± u(y)|

±

¤ Cρ ’d dD

s+d/ p

dy

s+d/ p

x’y

Bρ 2

D ± u(x) ’ D ± u

s+d/ p ’d/ p

¤ CdD ρ .

s+d/ p

x ’· 2 L p (D)

Raising both sides to the power p, integrating over D, and summing over all |±| = k gives

ρ ’d |u|W k+s (D) .

p sp+d p

|u ’ P|W k (D) ¤ C p dD

p p

A ¬nal application of dD /ρ ¤ 2γ and taking the pth root of both sides gives

|u ’ P|W p (D) ¤ CdD γ d/ p |u|W p (D) .

s

k k+s

Inserting this into the bound of Lemma 11.28 yields the result for m = 0. The gen-

eral case m > 0 follows from this, from (11.17), and from the relation |D ± u|W p (D)

k’|±|

¤ |u|W p (D) .

k

So far we have been concerned with polynomial approximation in Sobolev spaces over

small regions. We have not used the fact that our functions vanish on a discrete subset X at

198 Error estimates for radial basis function interpolation

all. We will employ this now in the way that we have already pointed out in the introductory

part of this section. Hence, our ¬rst estimate is

|u|W∞ (D) ¤ |u ’ Q k+1 u|W∞ (D) + |Q k+1 u|W∞ (D) . (11.18)

m m m

Our star-shaped domain D satis¬es a cone condition with radius ρ > 0 and angle ‘ =

2 arcsin [ρ/(2R)]. Thus, by Proposition 11.29 and the fact that the chunkiness parameter

satis¬es γ ¤ dD /r ¤ csc(‘/2), the ¬rst term can be bounded by

k+s’m’d/ p

|u ’ Q k+1 u|W∞ (D) ¤ CdD |u|W p (D)

m k+s

with a constant C depending only on k, d, p, and ‘. To bound the second term in (11.18)

we assume that the conditions in Proposition 11.7 are satis¬ed, so that by that proposition

we have the representation

N

±

a ± (x) p(x j ),

D p(x) = x ∈ D,

j

j=1

for all p ∈ πk (Rd ), with certain coef¬cients satisfying

|±|

N

2k 2

|a ± (x)| ¤ 2 , x ∈ D.

ρ sin ‘

j

j=1