Proof This simply follows from the de¬nition of the power function,

(») = »’μ ¤ »’μ

P inf inf

N ( )— N ( )—

,

μ∈span{ } μ∈span{ }

=P (»).

,

We now come back to our initial boundary-value problem, described at the beginning

of this section. Suppose that on the one hand condition (16.14) is satis¬ed. Then we know

that L is positive de¬nite, and the theory derived so far allows us to bound the error in the

interior simply by dropping all boundary functionals in the power function. On the other

hand, to derive estimates on the boundary we drop all interior functionals. As a consequence

the power function is reduced to the well-investigated power function for point evaluations.

Introducing the following additional notation,

= {δx1 , . . . , δxn },

1

= {δxn+1 , . . . , δx N },

2

= {δx1 —¦ L , . . . , δxn —¦ L} = —¦ L,

1 1

= {δxn+1 —¦ B, . . . , δx N —¦ B} = —¦ B,

2 2

we can give an estimate of the error bound in the interior:

|Lu(x) ’ Lsu, (x)| ¤ P (δx —¦ L) u

, N()

¤P (δx —¦ L) u

, N()

1

=P (δx —¦ L) u

, 1 —¦L N()

=P (δx ) u

L, N()

1

=P N ( ).

(x) u

L ,X 1

302 Generalized interpolation

The same is possible on the boundary, leading to

|Bu(x) ’ Bsu, (x)| ¤ P N ( ).

(x) u

B ,X 2

Let us demonstrate in more detail how this works in the case of positive de¬nite functions

in L 1 (Rd ). Since L is a differential operator and also since B contains in general at most

also certain derivatives, the ¬rst step is to answer the question when are the functionals

linearly independent. But this has already been done in Theorem 16.4. A consequence for

a linear differential operator is

Corollary 16.12 Let ∈ L 1 (Rd ) © C 2k (Rd ) be a positive de¬nite function. Suppose that

L : C k (Rd ) ’ C(Rd ) is a linear differential operator of order k, i.e. L = |±|¤k c± D ± with

c± ∈ C( ). Suppose further that either c± ≡ 0 or c± is nonzero everywhere and that not all

the c± vanish. Then L is also a positive de¬nite kernel.

Proof Let ± (1) , . . . , ± (M) be a numeration of all ± ∈ Nd with |±| ¤ k and c± nonzero, so

0

that the operator L takes the form L = M c D ± and the coef¬cient functions satisfy

()

˜

=1

c (x) = c±( ) (x) = 0 for all x ∈ and all 1 ¤ ¤ M. We want to apply Theorem 16.8. Thus

˜

we have to show that for arbitrary but distinct x1 , . . . , x N ∈ Rd and arbitrary b1 , . . . , b N ∈ R

the assumption

N N M

b j c (x j )D ± f (x j )

()

0= b j L f (x j ) = ˜ (16.15)

j=1 =1

j=1

for all f ∈ N (Rd ) leads to b1 = · · · = b N = 0. To this end, let us set y( j’1)M+ = x j ,

β (( j’1)M+ ) = ± ( ) , and d( j’1)M+ = b j c (x j ), each time for 1 ¤ j ¤ N and 1 ¤ ¤ M.

˜

Then assumption (16.15) becomes

NM

dk D β f (yk ) = 0

(k)

k=1

for all f ∈ N (Rd ). But the functionals »k = δ yk —¦ D β , 1 ¤ k ¤ N M, are pairwise dis-

(k)

tinct in the sense of Theorem 16.4. Hence dk = 0, 1 ¤ k ¤ N M, by that theorem. Since

c (x j ) = 0 this means that all the b j are zero.

˜

Thus the kernel is also positive de¬nite. But since it has the form

L

c± (x)cβ (y)(’1)|β| (D ±+β )(x ’ y)

y) = L x L y (x ’ y) =

L (x,

|±|,|β|¤k

it is no longer a translation-invariant kernel; the property of translation invariance is only

guaranteed if L has constant coef¬cients.

For the rest of this section we will assume Dirichlet boundary values, i.e. B = I , to make

life easier. Moreover, we restrict ourselves to polygonal regions, which is standard in ¬nite-

element theory. More general regions need interpolation by positive de¬nite functions on

(d ’ 1)-dimensional manifolds, which is the subject of the next chapter.

16.3 Solving PDEs by collocation 303

De¬nition 16.13 A bounded region ⊆ Rd is said to be a simple polygonal region if it is

the intersection of a ¬nite number of half spaces. A half space in Rd is a set Ha,b = {x ∈

Rd : a T x ¤ b} with a ∈ Rd \ {0} and b ∈ R. A region ⊆ Rd is said to be a polygonal

region if it is the union of a ¬nite number of simple polygonal regions.

A polygonal region is therefore bounded and obviously satis¬es an interior cone condition.

Moreover, its boundary is the union of a ¬nite number of (d ’ 2)-variate simple polygonal

regions.

To bound the error on the boundary, we need the following lemma.

∈ C(Rd ) is a positive de¬nite function and that X =

Lemma 16.14 Suppose that

{x1 , . . . , x N } ⊆ Rd is a set of pairwise distinct points. Suppose further that T : Rd ’ Rd

is a bijective af¬ne mapping, i.e. T x = Sx + c, x ∈ Rd , with an invertible matrix S ∈ Rd—d

and a constant c ∈ Rd . Then the the following relation for the power function holds:

=P x ∈ Rd .

P ,X (x) —¦S ’1 ,T (X ) (T x),

Here T (X ) denotes the set {T x1 , . . . , T x N }.

Proof Let u — (x; , X ), 1 ¤ j ¤ N , denote the cardinal functions with respect to and

j

X , i.e. u — (x; , X ) = A’1 R ,X (x) with A ,X = ( (x j ’ xk )) ∈ R N —N and R ,X (x) =

,X

( (x ’ x j )) ∈ R N . Set Z = T (X ), z = T (x). Because z j ’ z k = T x j ’ T xk = S(x j ’