also called Beppo Levi semi-norms)

D± u

p p

|u|W k ( L p ( ), k ∈ N0 , 1 ¤ p < ∞.

:=

)

p

|±|=k

The case p = ∞ is de¬ned in an obvious manner, replacing the sum by the maximum.

p

The full norm on W p ( ) is then given by summing up all semi-norms, i.e. u W k ( ) =

k

p

p

k

|u|W k ( ) . For an application to the compactly supported basis functions from Chapter 9

j=0 p

we also have to deal with fractional Sobolev spaces. One way to introduce them is by

interpolation theory. Here, we use the direct approach and de¬ne for 1 ¤ p < ∞, k ∈ N0 ,

and 0 < s < 1,

1/ p

|D ± u(x) ’ D ± u(y)| p

|u|W p ( := d xd y

k+s

) d+ ps

x’y

|±|=k 2

1/ p

p p

+ |u|W k+s ( .

u := u

k+s

Wp ( ) k

Wp ( ) )

p

Now suppose that u ∈ W p ( ) vanishes on X ⊆ and that k > d/ p ’ m, so that

k+s

W p ( ) ⊆ C ( ) by Sobolev™s embedding theorem. We wish to establish the following

k m

result:

k+s’|±|’d(1/ p’1/q)+

|u|Wqm ( ¤ ch X, |u|W p ( ) , (11.13)

k+s

)

which can be used immediately for our radial basis function interpolation process. Since the

proof of (11.13) involves a certain technicality, we will now outline its ideas for the special

case p = q = 2. The ¬rst step is to cover by “nice” local patches D of diameter O(h X, ).

The term “nice” will be explained very soon. Then on each patch D we approximate u by

a polynomial p ∈ πk (Rd ) that is an averaged Taylor polynomial:

|u|W2m (D) ¤ |u ’ p|W2m (D) + | p|W2m (D) . (11.14)

11.6 Sobolev bounds for functions with scattered zeros 195

The derivatives D ± p, |±| = m, of p can be expressed using Proposition 11.7 as

N N

±

a (±) (x) p(x j ) a (±) (x)[ p(x j ) ’ u(x j )],

D p(x) = =

j j

j=1 j=1

since u|X = 0. Moreover, we know that the coef¬cient vector in this representation can

’|±|

be bounded by Ch X, with a constant that depends only on the region geometry and

the polynomial degree (and the constant factor between the local cone radius and h X, ).

Inserting this into (11.14) yields

’m+d/2

|u|W2m (D) ¤ |u ’ p|W2m (D) + Ch X, |u ’ p| L ∞ (D) ,

d/2

where the additional h X, comes from the volume of D. Hence we have reduced everything

to a local polynomial approximation problem. When this is solved, the local estimates are

put together to form a global one. If the patches do not overlap too much, the sum of the

local Sobolev norms is equivalent to the Sobolev norm on . We now start the detailed

discussion.

De¬nition 11.25 A domain D is said to be star-shaped with respect to a ball B(xc , ρ) :=

{x ∈ Rd : x ’ xc 2 ¤ ρ} if, for every x ∈ D, the closed convex hull of {x} ∪ B is contained

in D. If D is bounded then the chunkiness parameter γ is de¬ned to be the ratio of the

diameter dD of D to the radius ρmax of the largest ball relative to which D is star-shaped.

A bounded domain D is contained in a ball B(xc , R). Throughout the rest of this section

we want to assume that ρmax /2 ¤ ρ ¤ ρmax , so that we have the obvious chain of inequalities

ρmax /2 ¤ ρ ¤ ρmax ¤ d D ¤ 2R. Hence, the chunkiness parameter satis¬es

1 dD dD 2R

¤ ¤γ = ¤ . (11.15)

ρmax ρ

2 2ρ

Such domains satisfy a simple interior cone condition.

Proposition 11.26 If D is bounded, star-shaped with respect to B(xc , ρ), and con-

tained in B(xc , R) then D satis¬es an interior cone condition with radius ρ and angle

‘ = 2 arcsin [ρ/(2R)].

Proof It is easy to check that when x ∈ B(xc , ρ) the cone condition is satis¬ed if the central

axis of the cone is directed along a diameter of the ball B(xc , ρ). If x is outside that ball

then we consider the convex hull of x and the intersection of the sphere S(x, x ’ xc 2 ) =

{y ∈ Rd : y ’ x 2 = xc ’ x 2 } with B(xc , ρ). This is a cone and, because D is star-

shaped with respect to B(xc , ρ), it is contained in D. Its radius is the distance from x to

xc . To ¬nd its angle ‘, we consider a triangle formed by x, xc , and any point on y in the

intersection of S(x, x ’ xc 2 ) and the sphere S(xc , ρ). This is an isosceles triangle, since

xc ’ x 2 = y ’ x 2 . The angle ∠xc x y = ‘; the side opposite this angle has length ρ.

A little trigonometry then gives us xc ’ x 2 sin(‘/2) = ρ/2. Consequently, we have ‘ =

2 arcsin [ρ/(2 xc ’ x 2 )]. Moreover, since D ⊆ B(xc , R), we also have xc ’ x 2 ¤ R.

196 Error estimates for radial basis function interpolation

Thus ‘ ≥ 2 arcsin [ρ/(2R)]. Finally, ρ ¤ x ’ xc 2 implies that the cone with vertex x,

axis along xc ’ x, and angle ‘ = 2 arcsin [ρ/(2R)] is contained in D.

After setting up the geometry of our local patches we introduce the approximating poly-

nomial. As stated before, this is an averaged Taylor polynomial:

1

D ± u(y)(x ’ y)± φρ (y)dy,

Q k u(x) :=

±! Bρ

|±|<k

∞

where φ ∈ C0 (Rd ) is chosen such that φρ = ρ ’d φ(·/ρ) is supported in Bρ = B(0, ρ) and

forms an approximation of the identity.

Our ¬rst result deals with the case of integer-order Sobolev spaces. We will omit the

proof here; it can be found in the book [31] of Brenner and Scott.

Proposition 11.27 Let 1 < p < ∞ and k > d/ p or p = 1 and k ≥ d/ p. Then there exists

a constant C > 0 depending only on k, d, p such that, for every u ∈ W p (D),

k

k’d/ p

u ’ Qk u ¤ C(1 + γ )d dD |u|W p (D) .

k

L ∞ (D)

Here γ denotes the chunkiness parameter for D.

k+s

We are now going to extend this result to fractional-order Sobolev spaces W p (D). To

k

this end, we ¬rst investigate the action of Q k+1 on a function from W p (D), which is at least