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35
The Structure of Finitely Generated ShiftInvariant Spaces in ...
, 1992
"... : 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 ..."
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Cited by 99 (20 self)
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: 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., shiftinvariant 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 shiftinvariant 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...
On linear independence of integer translates of a finite number of functions
 Proc. Edinburgh Math. Soc
, 1992
"... We investigate linear independence of integer translates of a finite number of compactly 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 som ..."
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Cited by 76 (32 self)
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We investigate linear independence of integer translates of a finite number of compactly 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 pnorms in terms of the L pnorm of the linear combination of integer translates of the basis functions which uses these coefficients. In both cases we give necessary 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.
Stability and linear independence associated with wavelet decompositions
 Proc. Amer. Math. Soc
, 1993
"... Wavelet decompositions are based on basis functions satisfying refinement equations. The stability, linear independence and orthogonality of the integer translates of basis functions play an essential role in the study of wavelets. In this paper we characterize these properties in terms of the mask ..."
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Cited by 59 (14 self)
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Wavelet decompositions are based on basis functions satisfying refinement equations. The stability, linear independence and orthogonality of the integer translates of basis functions 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 in Lp spaces
 Adv. Comput. Math
, 1995
"... 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 Lpconvergence of a subdivision scheme in terms of the pnorm joint spectral radius of two ..."
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Cited by 47 (21 self)
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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 Lpconvergence of a subdivision scheme in terms of the pnorm 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.
Local Decomposition of Refinable Spaces and Wavelets
, 1996
"... 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 o ..."
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Cited by 33 (8 self)
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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 nonorthogonal but in a certain sense stable and even local decompositions of nested spaces and to develop tools which are not necessarily confined to ...
Smooth Refinable Functions Provide Good Approximation Orders
, 1995
"... We apply the general theory of approximation orders of shiftinvariant spaces of [BDR13] to the special case when the finitely many generators \Phi ae L 2 (IR d ) of the underlying space S satisfy an Nscale relation (i.e., they form a "father wavelet" set). We show that the approximation orders ..."
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Cited by 29 (10 self)
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We apply the general theory of approximation orders of shiftinvariant spaces of [BDR13] to the special case when the finitely many generators \Phi ae L 2 (IR d ) of the underlying space S satisfy an Nscale relation (i.e., they form a "father wavelet" set). We show that the approximation orders provided by such finitely generated shiftinvariant 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 shiftinvariant 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 sideconditions are required if the spatial dimension is d = 1, and the functions in \Phi are compactly ...
Characterizations of Linear Independence and Stability of the Shifts of a Univariate Refinable Function in Terms of Its Refinement Mask
, 1992
"... : 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. ..."
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Cited by 19 (6 self)
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: 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, shiftinvariant spaces, refinement equation, stability, linear independence. Author's affiliation and address: Computer Science Department University of WisconsinMadison 1210 W. Dayton St. Madison WI 53706 email: amos@cs.wisc.edu Supported in part by the United States Army (Contract DAAL03G900090) and by the National Science Foundation (grants DMS9000053 and DMS9102857). 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 ...
Approximation orders of FSI spaces in ...
, 1996
"... A second look at the authors' ([BDR1], [BDR2]) characterization of the approximation order of a Finitely generated ShiftInvariant 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 ..."
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Cited by 18 (5 self)
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A second look at the authors' ([BDR1], [BDR2]) characterization of the approximation order of a Finitely generated ShiftInvariant 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 1periodizations 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]).
Affine frame decompositions and shiftinvariant spaces
 Appl. Comput. Harmon. Anal
"... In this paper, we show that the property of tight affine frame decomposition of functions in L 2 can be extended in a stable way to functions in Sobolev spaces when the generators of the tight affine frames satisfy certain mild regularity and vanishing moment conditions. Applying the affine frame op ..."
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Cited by 13 (9 self)
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In this paper, we show that the property of tight affine frame decomposition of functions in L 2 can be extended in a stable way to functions in Sobolev spaces when the generators of the tight affine frames satisfy certain mild regularity and vanishing moment conditions. Applying the affine frame operators Qj on jth levels to any function f in a Sobolev space reveals the detailed information Qjf of f in such tight affine decompositions. We also study certain basic properties of the range of the affine frame operators Qj such as the topological property of closedness and the notion of angles between the ranges for different levels, and thus establishing some interesting connection to (tight) frames of shiftinvariant spaces. 1.