This together with u|X = 0 allows us to estimate for |±| = m

N

|D ± (Q k+1 u)(x)| ¤ |a ± (x)||u(x j ) ’ Q k+1 u(x j )|

j

j=1

¤ Cρ ’m dD

k+s’d/ p

|u|W p (D)

k+s

k+s’m’d/ p

¤ CdD |u|W p (D) ,

k+s

with a generic constant C > 0 that depends only on d, p, k, m, and ‘. To derive the last

estimate we have once again used the upper bound dD /ρ ¤ csc(‘/2). Putting these two

bounds together gives the ¬rst part of the following proposition.

Proposition 11.30 Let k be a positive integer, 1 ¤ p < ∞, 0 < s ¤ 1, 1 ¤ q ¤ ∞, and let

m ∈ N0 satisfy k > m + d/ p, or, for p = 1, k ≥ m + d. Also, let X ⊆ D be a discrete set

that satis¬es the two conditions in Proposition 11.7 with an h > 0. If u ∈ W p (D) satis¬es

k+s

u| X = 0 then

k+s’m+d(1/q’1/ p)

|u|Wqm (D) ¤ CdD |u|W p (D) .

k+s

Here, the constant C depends only on k, d, p, m, and the angle ‘ corresponding to the cone

condition that D satis¬es.

Proof The remaining case q < ∞ follows from

d/q

|u|Wqm (D) ¤ C vol(D)1/q |u|W∞ (D) = CdD |u|W∞ (D) ,

m m

11.6 Sobolev bounds for functions with scattered zeros 199

where the ¬rst C is #{± ∈ Nd : |±| = m} = O(m d’1 ) and the second involves also the

0

volume of the unit ball in Rd .

This concludes our local estimates. It is important to notice that the constants depend on

the local domain D only via the angle ‘ and the radius ρ of the cone condition.

The next step is to cover our global region , which is supposed to be bounded and to

satisfy a cone condition with radius r and angle θ, using domains that are star-shaped. To

this end, we introduce the following quantities. Let h = h X, be the ¬ll distance and

sin θ

‘ := 2 arcsin ,

4(1 + sin θ)

sin θ sin ‘

Q(k, θ ) := 2 ,

8k (1 + sin θ )(1 + sin ‘)

R := Q(k, θ )’1 h,

sin θ

ρ := R.

2(1 + sin θ )

With these settings we de¬ne the sets

√

Tρ := t ∈ (2ρ/ d)Zd : B(t, ρ) ⊆

and

Dt = {x ∈ : co({x} ∪ B(t, ρ)) ⊆ © B(t, R)}, t ∈ Tρ ,

where co(A) denotes the closed convex hull of the set A.

Lemma 11.31 With the quantities just introduced, suppose that the ¬ll distance h = h X,

satis¬es h ¤ Q(k, θ)r . Then the following hold true:

(1) each Dt is star-shaped with respect to the ball B(t, ρ) and satis¬es B(t, ρ) ⊆ Dt ⊆ © B(t, R);

(2) each Dt satis¬es a cone condition with angle ‘ and radius ρ;

= t∈Tρ Dt and dDt ¤ 2R = 2h/Q(k, θ);

(3)

t∈Tρ χDt ¤ M1 ;

(4)

(5) #Tρ ¤ M2 ρ ’d .

Here χ B denotes the characteristic function of the set B and M1 , M2 are constants depending

only on k, θ, d.

Proof Obviously the ¬rst property is automatically satis¬ed for all ρ > 0. Hence, by

Proposition 11.26, Dt satis¬es a cone condition with radius ρ and angle

ρ R sin θ 1

= 2 arcsin = ‘.

2 arcsin

2(1 + sin θ) 2R

2R

Moreover, its diameter is bounded by dDt ¤ 2R = 2h/Q(k, θ ). Next note that our assump-

tion on h gives R ¤ r , so that also satis¬es a cone condition with radius R and angle θ.

Hence, if an arbitrary x ∈ is given then we can ¬nd a cone C(x) = C(x, ξ, θ, R) ⊆ . The

200 Error estimates for radial basis function interpolation

de¬nition of ρ and Lemma 3.7 ensure that the ball B(y, 2ρ) centered at y = x + (2ρ/sin θ)ξ

√

is contained in . For y we can choose a point t ∈ (2ρ/ d)Zd with y ’ t 2 ¤ ρ which

gives that the ball B(t, ρ) ⊆ B(y, 2ρ) is also contained in . Hence t ∈ Tρ and, since C(x)

is convex and since x ’ t 2 ¤ R, we additionally have co({x} ∪ B(t, ρ)) ⊆ © B(t, R),

so that x ∈ Dt . This shows the third property.

For the fourth property note that Dt ⊆ B(t, R) is contained in the cube W (t, R) and that

this cube contains at most

√ d

2 d(1 + sin θ)

+1

M1 :=

sin θ

√

points from (2ρ/ d)Zd . The last property is justi¬ed in the same way, since is

bounded.

Now that we have the local sets we can formulate and proof our main result of this section.

Theorem 11.32 Suppose is bounded and satis¬es an interior cone condition. Let k be a

positive integer, 0 < s ¤ 1, 1 ¤ p < ∞, 1 ¤ q ¤ ∞, and let m ∈ N0 satisfy k > m + d/ p,

for p > 1 or, for p = 1, k ≥ m + d. Also, let X ⊆ be a discrete set with mesh norm h

satisfying h ¤ Q(k, θ )r . If u ∈ W p ( ) satis¬es u| X = 0 then

k+s

|u|Wqm ( ¤ Ch k+s’m’d(1/ p’1/q)+ |u|W p ( ) , (11.19)

k+s

)

where (x)+ = x if x ≥ 0 and is 0 otherwise.

Proof We will use the notation introduced in the paragraph before Lemma 11.31. First of

all note that, since h ¤ Q(k, θ )r , Lemma 11.31 is applicable. Furthermore, our de¬nition

of ρ, R, and Q(k, θ ) establish

ρ sin ‘

h= ,

+ sin ‘)

4k 2 (1

which allows us to employ Proposition 11.30. The lemma and proposition just mentioned

immediately establish the result in the case q = ∞. For 1 ¤ q < ∞, however, the decom-

position of implies that we have

|D ± u(x)|q d x

q

|u|Wqm ( =

)

|±|=m

|D ± u(x)|q d x =

q

¤ |u|Wqm (Dt )