ńņš. 25 |

integrable functions with respect to a certain measure. Since this is standard again, we omit

the details here and proceed by stating those results that we will need later on.

be a locally compact metric space. If Ī» is a linear and

Theorem 5.33 (Riesz) Let

continuous functional on C0 ( ), which is nonnegative, meaning that Ī»( f ) ā„ 0 for all f ā

C0 ( ) with f ā„ 0, then there exists a nonnegative Borel measure Ī¼ on such that

Ī»( f ) = f (x)dĪ¼(x)

for all f ā C0 ( ). If possesses a countable basis then the measure Ī¼ is uniquely deter-

mined.

A metric space possesses a countable basis if there exists a sequence {U j } jāN of open

sets such that each open set U is the union of some of these sets.

Theorem 5.34 (Helly) Let {Ī½k } be a sequence of (signed) Borel measures on the compact

metric space of uniformly bounded total mass. Then there exists a subsequence {Ī½k j } and

a Borel measure Ī½ on such that

lim f (x)dĪ½k j (x)

jā’ā

exists for all f ā C( ) and equals f (x)dĪ½(x).

5.3 Measure theory 63

Finally, we need to know a result on the uniqueness of measures.

is a metric space and Ī¼ and Ī½ are

Theorem 5.35 (Uniqueness theorem) Suppose that

two ļ¬nite Borel measures on . If

f dĪ¼ = f dĪ½

for all continuous and bounded functions f , then Ī¼ ā” Ī½.

In this book we are mainly confronted with situations where is Rd , [0, 1], or [0, ā)

endowed with the induced topology. These sets are obviously metric (hence Hausdorff ),

locally compact, and possess a countable basis. Moreover, they are complete with respect

to their metric. Such spaces are sometimes called Polish spaces and have some remarkable

properties. For example, every ļ¬nite Borel measure is regular, meaning inner and outer

regular. This will reassure readers who might have wondered about regularity.

Finally, whenever we work on Rd and do not specify a Ļ -algebra or a measure, we will

assume tacitly that we are employing Borel sets and Lebesgue measure.

6

Positive deļ¬nite functions

With moving least squares we have encountered an efļ¬cient method for approximating

a multivariate function. The assumption that the weight function is continuous excludes

interpolation as a possible approximation process. If the weight function has a pole at the

origin, however, interpolation is possible. Nonetheless, moving least squares is a method

for approximation rather than for interpolation. The rest of this book is devoted to a very

promising method that allows interpolation in an arbitrary number of space dimensions and

for arbitrary choices of data sites.

6.1 Deļ¬nition and basic properties

From the Mairhuberā“Curtis theorem we know that there are no Haar spaces in the mul-

tivariate setting. If we still want to interpolate data values f 1 , . . . , f N at given data sites

X = {x1 , . . . , x N } ā Rd we have to take this into account. One simple way to do this is to

choose a ļ¬xed function : Rd ā’ R and to form the interpolant as

N

s f,X (x) = Ī± j (x ā’ x j ), (6.1)

j=1

where the coefļ¬cients {Ī± j } are determined by the interpolation conditions

s f,X (x j ) = f j , 1 ā¤ j ā¤ N. (6.2)

If we imagine for a moment that is a bump function with center at the origin, the shifts

(Ā· ā’ x j ) are functions that are centered at x j . Motivated by this point of view, we will

often call x j a center and X = {x1 , . . . , x N } a set of centers.

It would be nice if could be chosen for all kinds of data sets, meaning for any number

N and any possible combination X = {x1 , . . . , x N }. Obviously, the interpolation conditions

(6.2) imposed on a function s f,X of the form (6.1) are equivalent to asking for an invertible

interpolation matrix

:= ( (x j ā’ xk ))1ā¤ j,kā¤N .

A ,X

From the numerical point of view it is desirable to have more conditions on the matrix A ,X ,

64

6.1 Deļ¬nition and basic properties 65

for example that it is positive deļ¬nite. Later on, we will see that this requirement will turn

out quite naturally.

Deļ¬nition 6.1 A continuous function : Rd ā’ C is called positive semi-deļ¬nite if, for all

N ā N, all sets of pairwise distinct centers X = {x1 , . . . , x N } ā Rd , and all Ī± ā C N , the

quadratic form

N N

Ī± j Ī±k (x j ā’ xk )

j=1 k=1

is nonnegative. The function is called positive deļ¬nite if the quadratic form is positive

for all Ī± ā C N \ {0}.

Here, we have used a more general deļ¬nition for complex-valued functions. The reason

for this is that it allows us to use techniques such as Fourier transforms more naturally.

However, we will see that for even, real-valued functions it sufļ¬ces to investigate the

quadratic form only for real vectors Ī± ā R N .

The reader should note that we call a function positive deļ¬nite if the associated interpo-

lation matrices are positive deļ¬nite and positive semi-deļ¬nite if the associated matrices are

positive semi-deļ¬nite. This seems to be natural. Unfortunately, for historical reasons there

is an alternative terminology around in the literature: other authors call a function positive

deļ¬nite if the associated matrices are positive semi-deļ¬nite and strictly positive deļ¬nite if

the matrices are positive deļ¬nite. We do not follow this historical approach here, but the

reader should always keep this in mind when looking at other texts.

From Deļ¬nition 6.1 we can read off the elementary properties of a positive deļ¬nite

function.

Theorem 6.2 Suppose is a positive semi-deļ¬nite function. Then the following properties

are satisļ¬ed.

(1) (0) ā„ 0.

(2) (ā’x) = (x) for all x ā Rd .

is bounded, i.e. | (x)| ā¤ (0) for all x ā Rd .

(3)

(4) (0) = 0 if and only if ā” 0.

(5) If 1 , . . . , n are positive semi-deļ¬nite and c j ā„ 0, 1 ā¤ j ā¤ n, then := n c j j is also

j=1

positive semi-deļ¬nite. If one of the j is positive deļ¬nite and the corresponding c j is positive

then is also positive deļ¬nite.

(6) The product of two positive deļ¬nite functions is positive deļ¬nite.

Proof The ļ¬rst property follows by choosing N = 1 and Ī±1 = 1 in the deļ¬nition.

Next, setting N = 2, Ī±1 = 1, Ī±2 = c, x1 = 0, and x2 = x gives

(1 + |c|2 ) (0) + c (x) + c (ā’x) ā„ 0.

If we set c = 1 and c = i, respectively, this means that both (x) + (ā’x) and i[ (x) ā’

(ā’x)] must be real. This can only be satisļ¬ed if (x) = (ā’x), showing property (2).

66 Positive deļ¬nite functions

To prove the third property we take N = 2, Ī±1 = | (x)|, Ī±2 = ā’ (x), x1 = 0 and

x2 = x. Then the condition in the deļ¬nition and the fact that satisļ¬es (ā’x) = (x)

leads to

2| (x)|2 (0) ā’ 2| (x)|3 ā„ 0.

Property (4) follows immediately from the third. Property (5) is obvious. Property (6) is a

consequence of a theorem of Schur; we include the proof here. Since the interpolation matrix

T

A 2 ,X is positive deļ¬nite there exists a unitary matrix S ā C N Ć—N such that A 2 ,X = S DS ,

where D = diag{Ī»1 , . . . , Ī» N } is the diagonal matrix with the eigenvalues 0 < Ī»1 ā¤ Ā· Ā· Ā· ā¤

Ī» N of A 2 ,X as diagonal entries. This means that

N

ā’ xj) = s k s jk Ī»k .

ńņš. 25 |