which is symmetric and positive de¬nite whenever the functionals {» j } are linearly inde-

pendent over N ( ), because

y y

»k » j (x, y) = (»k (·, x), » j (·, y))N

x x

()

holds (see Theorem 16.7).

16.2 Hermite“Birkhoff interpolation

In this section we apply the general results of the last section to the speci¬c situation where

we are reconstructing a function from Hermite“Birkhoff data.

To be more precise, suppose that ± (1) , . . . , ± (N ) ∈ Nd are (not necessarily differ-

0

ent) multi-indices with |± ( j) | ¤ k. Suppose further that we are given certain points

x1 , . . . , x N ∈ from an open set ⊆ Rd . Then we form the functionals » j = δx j —¦ D ± ,

( j)

i.e. » j ( f ) = D ± f (x j ), 1 ¤ j ¤ N . To make these functionals pairwise different we

( j)

assume that for two different indices j = we have either x j = x or ± ( j) = ± ( ) .

If ∈ C 2k ( — ) is a positive de¬nite kernel on then we know from Theorem 10.45

that » j ∈ N ( )— and that » j has the Riesz representer

± ( j)

v j = D2 1 ¤ j ¤ N,

(·, x j ),

where the additional index 2 on the D-operator again denotes differentiation with respect

to the second argument. Hence, our interpolant takes the form

N

± ( j)

s= c j D2 (·, x j ) (16.5)

j=1

and the interpolation matrix has entries of the form

± ± () ( j)

(x , x j ).

D1 D2 (16.6)

This matrix is invertible whenever the functionals are linearly independent, and this depends

on how “rich” the native space of the underlying kernel is. Fortunately, all relevant basis

functions have a suf¬ciently rich native space. In accordance to the philosophy of this

chapter we give details only for positive de¬nite functions but it should become apparent

that the next theorem can also be adapted to the situation of conditionally positive de¬nite

functions.

Theorem 16.4 Suppose that ∈ L 1 (Rd ) © C 2k (Rd ) is positive de¬nite. If the functionals

» j := δx j —¦ D ± , 1 ¤ j ¤ N , with |± ( j) | ¤ k are pairwise distinct, meaning that ± ( j) = ± ( )

( j)

if x j = x for two different j = , then they are also linearly independent over N (Rd ).

16.2 Hermite“Birkhoff interpolation 293

N

c j » j = 0 on N (Rd ). This means

Proof Suppose we have real numbers c j such that j=1

in particular that

N N

y

0= cj»j = c j » j (· ’ y) .

j=1 j=1

N (Rd )— N (Rd )

By Theorem 10.12 we know that we can evaluate the last norm via Fourier transforms. To

this end we compute

» j (· ’ y) = (’1)|± |

D±

( j) ( j)

y

(· ’ x j )

and

(» j (· ’ y))§ (ω) = (’iω)± e’i x j ω (ω) = » j e’i y ω

T

( j) T

y y

(ω).

This yields

2 2

N N

= (2π )’d/2 e’i y ω

T

y y

cj»j (· ’ y) cj»j (ω)dω.

Rd

j=1 j=1

N (Rd )

Since is positive de¬nite there exists an open set U ⊆ Rd where (ω) > 0. Hence, we

must necessarily have

N

c j (iω)± ei x j ω = 0

T

( j)

(16.7)

j=1

for all ω with ’ω ∈ U .

Now we can proceed as in Lemma 6.7. Thus by analytic continuation we can see that

(16.7) is true for all ω ∈ Rd , not only for the ω with ’ω ∈ U . Then we can choose a test

function f ∈ S and get

§

N N

ixT ω

c j (iω)± e c j D ± f (· + x j )

( j) ( j)

0= f (ω) = (ω)

j

j=1 j=1

for all ω ∈ Rd , which implies

N

c j D ± f (x + x j ) = 0,

( j)

x ∈ Rd ,

j=1

and in particular, setting x = 0,

N

c j » j ( f ) = 0.

j=1

Finally, we have to choose the test function f appropriately. We let f 0 be a compactly

supported test function having support contained in the ball around zero with radius

0 < < min j=k x j ’ xk 2 and f 0 (x) = 1 if x 2 < /2. The latter means in particular

294 Generalized interpolation

that D ± f 0 (0) = 0 for all ± = 0. For 1 ¤ ¤ N we then de¬ne the function f = f to be

(x ’ x )±

()

f (x) = f 0 (x ’ x ), x ∈ Rd .

±( )!

Leibniz™ rule for multivariate functions now gives » j ( f ) = δ j, , showing that all coef¬cients

c j are zero.

Our approach differs crucially from the following naive, unsymmetric, approach. Since

at ¬rst sight it might seem undesirable that the functionals {» j } are applied twice to the

kernel to form the generalized interpolation matrix, one could be tempted to start instead

with a function of the form