MOMENT COMPUTATION IN SHIFT INVARIANT SPACES

An algorithm is given for the computation of moments of f E S, where S is either a principal h-shift invariant space or S is a finitely generated h-shift invariant space. An error estimate for the rate of convergence of our scheme is also presented. In so doing, we obtain a result for computing inner products in these spaces. As corollaries, we derive Marsden-type identities for principal h-shift invariant spaces and finitely generated h-shift invariant spaces. Applications to wavelet/multiwavelet spaces are presented.


Introduction
We consider the computation of the moment m(f) of a function f C L2(R). To this end, we project f into either an h-principal invariant subspace or a finitely generated shift invariant subspace. The approximation order and other characteristics of such 1Research partially supported by National Science Grant DMS9503282 and the Texas Regional Institute for Environmental Studies.
2Research partially supported by the National Science Foundation Grant DMS9503282.
spaces have been studied extensively in the fundamental paper [1] and again in [10,11,15].
Our main result deals with the computation of inner products in shift invariant spaces. The advantage of utilizing these spaces is the fact that we can often construct stable bases whose elements are integer translates of a compactly supported function or a finite number of compactly supported functions. Thus the computation of the inner product is easily implemented on a computer. As corollaries to our main result, we obtain as a special case the ability to compute moments of functions f0 E V 0. We then show how the process can be refined to obtain moments of the function fh Vh" The idea is to construct a sequence of shift invariant spaces V h approximating L2() in hopes of eventually approximating the moment of f G L2() by the moment of fh Vh" In the case where the vector that generates the finitely generated shift invariant space V 0 is refinable, we use a result of Cohen,Daubechies,and Plonka [5] in order to obtain an estimate of the error m(f)-As a consequence of our main result, we characterize a Marsden's identity for finitely generated shift invariant spaces. Recall Marsden's identity gives the explicit representation of x n in terms of B-splines [12]. A multivariate analog for box splines is given in [3].
The outline of this paper is as follows: In Section 2, we give basic definitions and elementary results necessary to the sequel. An inner product theorem and related corollaries concerning moment computation and Marsden's identity are given in Section 3. In addition, we give an error estirnate for the difference between the moment of f L2() and the moment of its orthogonal projection fh in a shift invariant subspace of L2(N). The final section contains moment recursion formulas for refinable functions and vectors as well as examples of illustrating our results.

Notation, Definitions, and Basic Results
In this section we introduce notation, definitions, and basic results used throughout the remainder of the paper. Let us begin by defining various types of shift invariant spaces.
Suppose V h is a linear space and h > 0. Then V h is said to be an h-shift invar- When h 1, we will suppress the h in the definitions above and refer to the spaces as shift invariant and principal shift invariant, respectively. As a matter of convention, we will denote an h-shift invariant space generated by 4) by Vh(). We will also be interested in shift invariant spaces generated by several functions. That is, we define a finitely generated shift invariant space V h by insisting that f e Vh::f(x (ak)Te(x-where now ()-(), and the a k E r. Such a space generated by b will be denoted by V h(b).
We readily observe the following properties: (2) If V is a shift invariant space generated by b, then V h is an h-shift invariant space generated by b().
We will say that the h-shift invariant space V h is of degree n and write deg(Vh) n if x k Vh, k 0,..., n, but x n + it Vh. Since the degree of a polynomial is invariant under dilation, we observe that deg(Vh) deg(V).
If deg(V())= n, we will call any identity of the form 0,..., n a Marsden's identity. The identity is easily generalized to the case where V is generated by b.
Suppose generates the principal shift invariant space V. In this paper, we always assume that is integrable. Recall that if is compactly supported and integrable, then is also in L2. Now we say a function * V is the dual of if Analogously, if vector b generates the space V, then we will define its dual b* V as the vector of functions that satisfy We will say is stable if ((. k), em(" J)) 5mbkj" I( + )1 : > 0 v e n and vector 4' is sable if he r r matrix ,I,:-(j())[j _ is positive definitive where () ( + 1( + 2).
Recall that if b is stable, then b* also generates a stable basis for V.
Suppose {(. -k)},k e i, -1,...,r forms a basis for Y. Then it also forms a basis for V n L2(). For convenience, we will use V to denote V g L2(R where Pk are r r matrices, then IIm_ 1 C V() is equivalent to the existence of solutions to two systems of equations involving the refinement mask of (see [5,10,15]). When is refinable in the sense of (1), we can obtain estimates on the accuracy of approximating moments of f E L2() using projections of f into Vh().

Main Results
The main goal of this paper is to provide an algorithm for computing moments in principal (or finitely generated) shift invariant spaces. Once the algorithm is in place, we will use it to attempt to approximate the moment m(f) of f L2(). In order to obtain formula for computing moments of f0-Prjyf and subsequent moments in refined spaces, we establish the following result.
Theorem 3.1: Suppose b is compactly supported, stable and that * is its dual.
Denote by V() the space generated by . Assume f U() satisfies the decay condi- (3) and g V() satisfies the growth condition Then (f g) E k e 7]( ak) T bk" Proof: Without loss of generality, assume that supp()-[0, L]. (By the support of a vector e Nr, we mean supp()-J 0supp(/).) First note that since is compactly supported, b* is of exponential decay. That is, for k 1,..., r: (x) <_ C1 ex (6) for some 7 > 0. k and k We shall now estimate the decay rate of a hi, k E 1_, and i-1,...,r. First we show that for sufficiently large k and constant C 2 > 0, we have ai <C2lkl (7) Since is compactly supported and stable, the set {*(. k)} k e 7] forms an unconditional basis for V( In an analogous manner, define f_ M and g_ M" Then (7), (8) and the fact that a-/3 > 1 we see that the second term in the above sum tends to 0 as M-oc so that we obtain the desired result. E! We obtain the following moment formulas as immediate corollaries of Theorem 3.1.
Corollary 3.2: Let V() be generated by a compactly supported and stable and let ek* be the dual of . Further assume that deg(V(qb))n, : and that f E V(b) satisfies the decay condition fl O,...,n, where c-/3 > l. Then Corollary 3.2 illustrates how we may compute the moments of order / or less of f E U(b). Note that in order to implement (11), we must have the coefficient vector c k' for xf. We will discuss a procedure for obtaining c k' f later in this section.
The following corollary describes how we can refine our procedure and obtain moments m(f) where f Vh(ek).  (9). Then m(f) h + 1 E (ak)T ck'" We continue by deriving explicit representations for ck'z.
Thus to compute the ck'f, we need only the moments of order less than/3 of the components of 0. We shall see in the final section of the paper that in the case of the paper that in the case where is refinable then this task can be performed recursively. Once we have these moments, we can see Corollary 3.2 or Corollary 3.3 to compute moments of functions in (finitely generated) principal shift invariant subspaces of L2(N). In addition Proposition 2.1 provides a means to estimate moments from these spaces should we intend to use them to approximate moments of functions in We conclude this section by noting that in light of (9) and (13)

Refinable Functions and Vectors
In the final section of the paper, we discuss various methods for computing the initial moments mf(j) as given in (13). One of the most popular ways to obtain classes of (finitely generated) principal shift invariant spaces is to use ideas from wavelet or multiwavelet theory (see [2,6] for wavelets, [8,9] for multiwavelets). The idea is to construct a nested ladder of 2 principal shift invariant subspaces of L (N). This ladder is constructed by finding a function (or a vector ) who along with its integer translates forms a Reisz basis for a space V 0. For k E , the space V k is formed using the translates (2kxn), n E l), of (2kx). Other requirements are made of the nested ladder to ensure existence of a wavelet. The property we are particularly interested in is the refinement property (1).
Our first results of this section shows that if the generator is refinable, then we need only compute f (x)dx and then this value to recursively generate all moments needed in (13). Proposition 4.1: Assume that/3 >_ 1, t3 -, and that f (x)dx 7 O. Furthermore suppose that there exists real numbers PO,"',PN so that N (x) 1 / 2 E Pk (2x k). It is possible to generalize this result to the vector case. To this end, we introduce some new notation. Let m/3() E r r be the vector whose components are given by mf(b)/-m/3(), I-1,...,r. In addition, we define the r r matrix P by N k=O where the P/, satisfy the refinement condition (1).  We will estimate the fl-1,2,3,4 moments of f(. ;1/2,1/2) using each of 2,'",5" In order to do so, we must calculate the 0 order moment for each scaling function. We can then use the recursion formula in Proposition 4.1 to compute the higher order moments that are used to form the ck'. The next step is to obtain the ak's in Corollary 3.3. We provide our results for h-2-J, where j-6,8, 10.
We have provided three different methods for obtaining the ak. In the first case, we simply sample f. In the second case, we approximate f by a piecewise quadratic polynomial and then use the precomputed c k'4 to form a Newton-Cotes type integration scheme. The third method for computing the a k uses the numerical integration from the prior method but uses a more sensitive approximation to f at the breakpoints x-0 and x-1.
Our results are given in the tables below. The numbers in parentheses represent the error between the approximation and the exact value. Note that we have used 2, 3, and 4 even in cases where Corollary 3.2 does not apply (that is, the order of the moment is larger than the degree of the space). The error in these cases is larger since x is not a member of the spaces generated by the corresponding scaling functions. In addition, since we must approximate the {ak} in some fashion, the errors are dependent on the function f. Since f is only C , it is nature that the C o function 2 does an adequate job approximating the moments. Since the support of 2 is less than that of any other scaling function we use, the expansions for fh consist of fewer terms. Thus, the computational cost of using 2 is less than that incurred by the other scaling functions.    We now consider an example illustrating our methods with finitely generated shift invariant spaces. To this end, we employ the scaling vector comprised of fractal interpolation functions given in [7]. We also note that in the case of these functions, a recursion formula for the moments exists (see [14]) and could be used in place of Proposition 4.2.
Example 4.4: We consider the closed linear space V 0 spanned by the fractal interpolation functions 1 and 2 as derived in [7]. We choose 1, 2 so that deg(Uo) 3.
We obtain the Vj spaces by taking the closed linear span of the set {2-u/2(2Jx-k), -1,2}k E " As in Example 4.3, we approximate moments of the beta distribution. Note that the accuracy is about the same as that of the Daubechies 2 function.
In the first table, the a k were function samples; in the second table, the a k were obtained using numerical integration; in the third table, the a k were obtained using adaptive numerical integration. The adaptive integration is not as effective here since one of the scaling functions has the same support as does f. The actual values for rn(f), j3-1,2, 3, 4 are given in the previous example.