Abstract not found

: A simple characterization is given of finitely generated subspaces of L 2 (IR d ) which are invariant under translation by any (multi)integer, and used to give conditions under which such a space has a particularly nice generating set, namely a basis, and, more than that, a basis with desirable properties, such as stability, orthogonality, or linear independence. The last property makes sense only for `local' spaces, i.e., shift-invariant spaces generated by finitely many compactly supported functions, and special attention is paid to such spaces. As an application, we prove that the approximation order provided by a given local space is already provided by the shift-invariant space generated by just one function, with this function constructible as a finite linear combination of the finite generating set for the whole space, hence compactly supported. This settles a question of some 20 years' standing. AMS (MOS) Subject Classifications: primary: 41A25, 41A63, 46C99; secondary: 4...

We investigate linear independence of integer translates of a finite number of com-pactly supported functions in two cases. In the first case there are no restrictions on the coefficients that may occur in dependence relations. In the second case the coefficient sequences are restricted to be in some ℓ p space (1 ≤ p ≤ ∞) and we are interested in bounding their ℓ p-norms in terms of the L p-norm of the linear combination of integer translates of the basis functions which uses these coefficients. In both cases we give nec-essary and sufficient conditions for linear independence of integer translates of the basis functions. Our characterization is based on a study of certain systems of linear partial difference and differential equations, which are of independent interest.

Wavelet decompositions are based on basis functions satisfying refinement equations. The stability, linear independence and orthogonality of the integer translates of basis func-tions play an essential role in the study of wavelets. In this paper we characterize these properties in terms of the mask sequence in the refinement equation satisfied by the basis function.

Subdivision schemes play an important role in computer graphics and wavelet analysis. In this paper we are mainly concerned with convergence of subdivision schemes in Lp spaces (1 ≤ p ≤ ∞). We characterize the Lp-convergence of a subdivision scheme in terms of the p-norm joint spectral radius of two matrices associated with the corresponding mask. We also discuss various properties of the limit function of a subdivision scheme, such as stability, linear independence, and smoothness.

A convenient setting for studying multiscale techniques in various areas of applications is usually based on a sequence of nested closed subspaces of some function space F which is often referred to as multiresolution analysis. The concept of wavelets is a prominent example where the practical use of such a multiresolution analysis relies on explicit representations of the orthogonal difference between any two subsequent spaces. However, many applications prohibit the employment of a multiresolution analysis based on translation invariant spaces on all of IR s , say. It is then usually difficult to compute orthogonal complements explicitly. Moreover, certain applications suggest using other types of complements, in particular, those corresponding to biorthogonal wavelets. The main objective of this paper is therefore to study possibly non-orthogonal but in a certain sense stable and even local decompositions of nested spaces and to develop tools which are not necessarily confined to ...

We apply the general theory of approximation orders of shift-invariant spaces of [BDR1-3] to the special case when the finitely many generators \Phi ae L 2 (IR d ) of the underlying space S satisfy an N-scale relation (i.e., they form a "father wavelet" set). We show that the approximation orders provided by such finitely generated shift-invariant spaces are bounded from below by the smoothness class of each / 2 S (in particular, each OE 2 \Phi), as well as by the decay rate of its Fourier transform. In fact, similar results are valid for refinable shift-invariant spaces that are not finitely generated. Specifically, it is shown that, under some mild technical conditions on the scaling functions \Phi, approximation order k is provided if either some / 2 S lies in the Sobolev space W k\Gamma1 2 , or its Fourier transform b /(w) decays near 1 like o(jwj 1\Gammak ). No technical side-conditions are required if the spatial dimension is d = 1, and the functions in \Phi are compactly ...

: Characterizations of the linear independence and stability properties of the integer translates of a compactly supported univariate refinable function in terms of its mask are established. The results extend analogous ones of Jia and Wang which were derived for dyadic refinements and finite masks. AMS (MOS) Subject Classifications: primary: 39B32, 41A15, 46C99; secondary: 42A99, 46E20. Key Words and phrases: wavelets, multiresolution, shift-invariant spaces, refinement equation, stability, linear independence. Author's affiliation and address: Computer Science Department University of Wisconsin-Madison 1210 W. Dayton St. Madison WI 53706 e-mail: amos@cs.wisc.edu Supported in part by the United States Army (Contract DAAL03-G-90-0090) and by the National Science Foundation (grants DMS-9000053 and DMS-9102857). Characterizations of linear independence and stability of the shifts of a univariate refinable function in terms of its refinement mask Amos Ron 1. The problem Let ...

A second look at the authors' ([BDR1], [BDR2]) characterization of the approximation order of a Finitely generated Shift-Invariant subspace S(#) of L 2 (IR )results in a more explicit formulation entirely in terms of the (Fourier transform of the) generators # # of the subspace. Further, when the generators satisfy a certain technical condition, then, under the mild assumption that the set of 1-periodizations of the generators is linearly independent, such a space is shown to provide approximation order k if and only if span{#(-j):|j| <k,## #} contains a # (necessarily unique) satisfying #(#)=# j # # for <k, # . The technical condition is satisfied, e.g., when the generators are O(|| -# ) at infinity for some #>k+ d. In the case of compactly supported generators, this recovers an earlier result of Jia ([J1], [J2]).

Let OE be a distribution solution of the two-scale difference equation (1). First the equivalence of local and global linear independence of the integer translates of OE is proved and a simple characterization for global linear independence of the integer translates of OE is given. Second a class of functions in V 1 such that their integer translates are locally or globally linearly independent is found. Key words: two-scale difference equation, global linear independence, local linear independence,B-spline,B-wavelet.** 1.Preliminary and Statement of Results. The objective of this context is to study local and global linear independence of the integer translates of a distribution solution of the two-scale equation. To this end, we introduce some notations and definitions. Let fc k g N k=0 be a sequence such that c 0 6= 0; c N 6= 0 and P N k=0 c k = 2: Let OE be a unique complex-valued compactly supported distribution to satisfy a two-scale difference equation 8 ? ? ! ? ? : OE(x)...