Proposition 11.6 Suppose that ⊆ Rd is bounded and satis¬es an interior cone condition

with radius r > 0 and angle θ . If p ∈ π (Rd ) and ± ∈ N0 is a multi-index for which |±| ¤

then

|±|

22

±

¤ L∞( ).

Dp p

L∞( )

r sin θ

Proof Obviously the result is true if ± = 0 or D ± p = 0. Hence let us assume that ∇ p

is not identically zero. The maximum of ∇ p(x) 2 over occurs at some point x M ∈ .

Obviously, the maximum is positive. Let · = ∇ p(x M )/ ∇ p(x M ) 2 . Because x M ∈ , the

cone condition, which holds for as well as , implies that x M is the vertex of a cone

C ⊆ having radius r , axis along a direction ξ , and angle θ . We may adjust the sign of

p so that · T ξ ≥ 0. By looking at the intersection of the cone C with a plane containing

ξ and ·, we see that there is a unit vector ζ pointing into the cone and satisfying · T ζ ≥

cos(π/2 ’ θ ) = sin θ. It follows that

‚p ‚p

∇ p(x M ) = (x M ) ¤ csc θ (x M ).

2

‚· ‚ζ

However, for t ∈ R, p(t) := p(x M + tζ ) is in π (R). In particular, it obeys the usual Bern-

˜

stein inequality on 0 ¤ t ¤ r :

22 22

| p (t)| ¤ max | p(t)| ¤ L∞( ).

˜ ˜ p

r t∈[0,r ] r

Since p (0) = (‚ p/‚ζ )(x M ), we have for all x ∈

˜

‚p 22

∇ p(x) ¤ ∇ p(x M ) ¤ csc θ (x M ) ¤ L∞( ).

p

2 2

‚ζ r sin θ

Noting that |(‚ p/‚x j )(x)| ¤ ∇ p(x) 2 and, keeping track of polynomial degrees as we

differentiate, we obtain the stated result.

With this result, Theorem 3.4 together with Theorem 3.8 immediately yields the global

version of a polynomial reproduction.

Proposition 11.7 Let p ∈ π (Rd ) and let be bounded, satisfying an interior cone condi-

tion with radius r > 0 and angle θ. Suppose that h > 0 and the set X = {x1 , . . . , x N } ⊆

satisfy

11.2 Error estimates in terms of the ¬ll distance 179

r sin θ

(1) h ¤ ,

4(1 + sin θ) 2

(2) for every B(x, h) ⊆ there is a center x j ∈ X © B(x, h);

then for any multi-index ± with |±| ¤ there exist real numbers a ± (x) such that

j

N

±

a ± (x) p(x j )

D p(x) = j

j=1

for all p ∈ π (Rd ). Moreover,

|±|

N

22

|a ± (x)| ¤2 .

r sin θ

j

j=1

As pointed out after the proof of Theorem 3.8, the second condition is automatically

satis¬ed if h is the ¬ll distance h X, . Using again the fact that a cone satis¬es a cone

condition itself, we can proceed as in Section 3.3 to derive the following local version.

Theorem 11.8 Suppose that ⊆ Rd is bounded and satis¬es an interior cone condition.

(±) (±)

Let ∈ N0 and ± ∈ Nd with |±| ¤ . Then there exist constants h 0 , c1 , c2 > 0 such

0

that for all X = {x1 , . . . , x N } ⊆ with h X, ¤ h 0 and every x ∈ there exist numbers

(±) (±)

u 1 (x), . . . , u N (x) with

±

N (±)

j=1 u j (x) p(x j ) = D p(x) for all p ∈ π (Rd ),

(1)

(±) ’|±|

N (±)

j=1 |u j (x)| ¤ c1 h X, ,

(2)

u (±) (x) = 0, if x ’ x j 2 > c2 h X, .

(±)

(3) j

Note that the construction ensures also the following important property. Given x ∈ ,

at most those u (±) (x) that belong to a center x j in the cone associated with x are nonzero.

j

Hence all line segments that connect one of these xi with either x or another x j are also

contained in . This allows us to apply Taylor™s formula later on.

Our ¬rst main result deals with (conditionally) positive de¬nite functions. After it, we will

also deal with (conditionally) positive de¬nite kernels. To treat the case of a (conditionally)

positive de¬nite function we have to remark that an even function : Rd ’ R that is in

C k (Rd ) gives rise to a symmetric kernel (· ’ ·) that is in C 2 k/2 (Rd — Rd ). Moreover, we

do not have to restrict ourselves to open regions , even in the case of derivatives, because

any function f ∈ N ( ) is the restriction of a function from N (Rd ) ⊆ C k/2 (Rd ).

Theorem 11.9 Suppose that ∈ C k (Rd ) is conditionally positive de¬nite of order m. Sup-

pose further that ⊆ Rd is bounded and satis¬es an interior cone condition. Fix ≥ m ’ 1.

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

0

function can be bounded:

2

P (±) (x) ¤ |D 2± (0) ’ D 2± p(0)|

,X

’|±|

D± ’ D± p

(±)

+ 2c1 h X, (±)

L ∞ (B(0, c2 h X, ))

’2|±|

(±)

+ [c1 ]2 h X, ’p L ∞ (B(0, 2c2 h X, )) , (11.5)

(±)

180 Error estimates for radial basis function interpolation

(±) (±)

where p is an arbitrary polynomial from π (Rd ) and the constants h 0 , c1 , c2 come from

Theorem 11.8.

Proof Let us introduce the notation

:= |D 2± (0) ’ D 2± p(0)|,

0

:= D ± ’ D± p (±)

1 L ∞ (B(0, c2 h X, ))

’p L ∞ (B(0, 2c2 h X, )) .

:= (±)

2

The polynomial reproduction property of the functions u (±) from Theorem 11.8 gives

(±) ± |±| ±

N

j=1 u j D p(x ’ x j ) = (’1) D p(0) and, when applied twice,

N N

u i(±) (x)u (±) (x) p(xi ’ x j ) = (’1)|±| u i(±) (x)D ± p(xi ’ x)

j

i, j,=1 i=1

|±|

= (’1) D 2± p(0).

Hence, if we rewrite the quadratic form Q with replaced by p, we ¬nd that Q p (u (±) ) = 0.

Thus we can bound the power function: