P (±) (x) ¤ Q (u (±) ) ’ Q p (u (±) ) = Q (±)

’ p (u )

,X

N N

u (±) (x) u i(±) (x) u (±) (x)

¤ + +

0 1 2

j j

j=1 i, j=1

’|±| 2 ’2|±|

(±) (±)

¤ + 2c1 h X, + c1 h X, 2.

0 1

This holds for any p ∈ π (Rd ). In the preceding estimates we have in particular employed

the other two features of the coef¬cients.

Note that if one is interested only in function values, the error estimate of the last theorem

becomes

2

(0)

P 2 ,X (x) ¤ 1 + c1 ’p L ∞ (B(0, 2c2 h X, )) .

(0)

Furthermore, in the case of a radial function = φ( · 2 ) one can use univariate polyno-

mials p ∈ πn (R) as multivariate polynomials via p( · 2 ) ∈ π2n (R ), which leads to the

2 d

estimates

2

(0)

P 2 ,X (x) ¤ 1 + c1 |φ(r ) ’ p(r 2 )|

max

(0)

0¤r ¤2c2 h X,

√

2

(0)

= 1 + c1 |φ( s) ’ p(s)|.

max (11.6)

(0)

0¤s¤[2c2 ]2 h 2

X,

Theorem 11.9 allows us to state our ¬rst generic error estimate.

De¬nition 11.10 The space Cν (Rd ) is de¬ned to consist of all functions f ∈ C k (Rd ) whose

k

derivatives of order k satisfy D ± f (x) = O( x ν ) for x 2 ’ 0.

2

11.2 Error estimates in terms of the ¬ll distance 181

Theorem 11.11 Suppose that ∈ Cν (Rd ) is conditionally positive de¬nite of order m.

k

Suppose further that ⊆ R is bounded and satis¬es an interior cone condition. For

d

± ∈ Nd with |±| ¤ k/2 and X = {x1 , . . . , x N } ⊆ satisfying h X, ¤ h 0 we have the error

0

bound

D ± f ’ D ± s f,X

(k+ν)/2’|±|

¤ Ch X, | f |N ( ).

L∞( )

Proof We ¬x ≥ max{m ’ 1, k ’ 1} and take p as the Taylor polynomial of of degree

k ’ 1, i.e. p(x) = |β|<k D β (0)x β /β!. Since ∈ Cν (Rd ), we immediately get for any

k

|γ | ¤ k

|D β+γ (ξ )| β k’|γ |+ν

γ γ

|D (x) ’ D p(x)| ¤ |x | ¤ Ch X, ,

β!

|β|=k’|γ |

provided that x 2 ¤ ch X, . Inserting the corresponding results for γ = 0, ±, 2± into (11.5)

gives the desired result.

Remark 11.12 It is worthwhile to note that the error estimate of the preceeding theorem

holds for every h ¤ h 0 which satis¬es the condition that every ball B(x, h) ⊆ contains

at least one point from X . Moreover, h 0 = r/C2 with C2 from Theorem 3.14. Finally, the

constant C depends on only via its cone condition angle θ .

We end this section by discussing the kernel case.

Theorem 11.13 Let ⊆ Rd be open and bounded, satisfying an interior cone condition.

Suppose that ∈ C 2k ( — ) is conditionally positive de¬nite with respect to πm’1 (Rd ).

Denote the interpolant to f ∈ N ( ) that is based on the πm’1 (Rd )-unisolvent set X =

{x1 , . . . , x N } by s f,X . Fix ± ∈ Nd with |±| ¤ k. Then there exist constants h 0 , C > 0 such

0

that

|D ± f (x) ’ D ± s f,X (x)| ¤ CC (x)1/2 h X,

k’|±|

| f |N ( ), x∈ ,

if h X, ¤ h 0 . The number C (x) is de¬ned by

β ν

D1 D2 (z, w) .

C (x) := max max

β,ν∈Nd (±)

z,w∈ ©B(x, c2 h X, )

0

|β|+|ν|=2k

and the constant C is independent of x, f , and .

Proof We will make use in the following of two Taylor expansions. In both cases we keep

the ¬rst argument w ∈ ¬xed and expand the function with respect to its second argument

around w. Moreover, we have to ensure that the line segment between w ∈ and z ∈ is

also contained in . Fix ν ∈ Nd with |ν| ¤ k. The ¬rst expansion is

0

β ν

D2 D1 (w, w)

ν

(z ’ w)β + R(w, z, ν),

(w, z) =

D1

β!

|β|<2k’|ν|

182 Error estimates for radial basis function interpolation

with remainder

β ν ν

D2 D1 (w, ξw,z )

(z ’ w)β .

R(w, z, ν) :=

β!

|β|=2k’|ν|

ν

ξw,z is a point on the line segment between w and z. Similarly, we have the second

Here,

expansion,

β+ν

(w, w)

D2

ν

(z ’ w)β + S(w, z, ν).

(w, z) =