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63
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 101 (21 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...
Approximation properties of multivariate wavelets
 Math. Comp
, 1998
"... Abstract. Wavelets are generated from refinable functions by using multiresolution analysis. In this paper we investigate the approximation properties of multivariate refinable functions. We give a characterization for the approximation order provided by a refinable function in terms of the order of ..."
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Cited by 66 (10 self)
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Abstract. Wavelets are generated from refinable functions by using multiresolution analysis. In this paper we investigate the approximation properties of multivariate refinable functions. We give a characterization for the approximation order provided by a refinable function in terms of the order of the sum rules satisfied by the refinement mask. We connect the approximation properties of a refinable function with the spectral properties of the corresponding subdivision and transition operators. Finally, we demonstrate that a refinable function in W k−1 1 (Rs) provides approximation order k. 1.
Multivariate refinement equations and convergence of subdivision schemes
 SIAM J. Math. Anal
, 1998
"... Abstract. Refinement equations play an important role in computer graphics and wavelet analysis. In this paper we investigate multivariate refinement equations associated with a dilation matrix and a finitely supported refinement mask. We characterize the Lpconvergence of a subdivision scheme in te ..."
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Cited by 61 (45 self)
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Abstract. Refinement equations play an important role in computer graphics and wavelet analysis. In this paper we investigate multivariate refinement equations associated with a dilation matrix and a finitely supported refinement mask. We characterize the Lpconvergence of a subdivision scheme in terms of the pnorm joint spectral radius of a collection of matrices associated with the refinement mask. In particular, the 2norm joint spectral radius can be easily computed by calculating the eigenvalues of a certain linear operator on a finite dimensional linear space. Examples are provided to illustrate the general theory.
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 60 (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.
Biorthogonal Wavelet Expansions
 Constr. Approx
"... This paper is concerned with developing conditions on a given finite collection of compactly supported algebraically linearly independent refinable functions that insure the existence of biorthogonal systems of refinable functions with similar properties. In particular we address the close connectio ..."
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Cited by 50 (7 self)
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This paper is concerned with developing conditions on a given finite collection of compactly supported algebraically linearly independent refinable functions that insure the existence of biorthogonal systems of refinable functions with similar properties. In particular we address the close connection of this issue with stationary subdivision schemes. Key Words: Finiteley generated shiftinvariant spaces, stationary subdivision schemes, matrix refinement relations, biorthogonal wavelets. AMS Subject Classification: 39B62, 41A63 1 Introduction During the past few years the construction of multivariate wavelets has received considerable attention. It is quite apparent that multivariate wavelets with good localazition properties in frequency and spatial domains which constitute an orthonormal basis of L 2 (IR s ) are hard to realize. On the other hand, it turns out that in many applications orthogonality is not really important whereas locality, in particular, compact support is very...
Multiresolution and wavelets
 Proc. Edinburgh Math. Soc
, 1994
"... Multiresolution is investigated on the basis of shiftinvariant spaces. Given a finitely generated shiftinvariant subspace S of L2(IR d), let Sk be the 2 kdilate of S (k ∈ Z). A necessary and sufficient condition is given for the sequence {Sk}k ∈ Z to form a multiresolution of L2(IR d). A general ..."
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Cited by 48 (24 self)
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Multiresolution is investigated on the basis of shiftinvariant spaces. Given a finitely generated shiftinvariant subspace S of L2(IR d), let Sk be the 2 kdilate of S (k ∈ Z). A necessary and sufficient condition is given for the sequence {Sk}k ∈ Z to form a multiresolution of L2(IR d). A general construction of orthogonal wavelets is given, but such wavelets might not have certain desirable properties. With the aid of the general theory of vector fields on spheres, it is demonstrated that the intrinsic properties of the scaling function must be used in constructing orthogonal wavelets with a certain decay rate. When the scaling function is skewsymmetric about some point, orthogonal wavelets and prewavelets are constructed in such a way that they possess certain attractive properties. Several examples are provided to illustrate the general theory.
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.
Characterization of smoothness of multivariate refinable functions in Sobolev spaces
 Trans. Amer. Math. Soc
, 1999
"... Abstract. Wavelets are generated from refinable functions by using multiresolution analysis. In this paper we investigate the smoothness properties of multivariate refinable functions in Sobolev spaces. We characterize the optimal smoothness of a multivariate refinable function in terms of the spect ..."
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Cited by 45 (4 self)
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Abstract. Wavelets are generated from refinable functions by using multiresolution analysis. In this paper we investigate the smoothness properties of multivariate refinable functions in Sobolev spaces. We characterize the optimal smoothness of a multivariate refinable function in terms of the spectral radius of the corresponding transition operator restricted to a suitable finite dimensional invariant subspace. Several examples are provided to illustrate the general theory. 1.
Vector subdivision schemes and multiple wavelets
 Math. Comput
, 1998
"... Abstract. We consider solutions of a system of refinement equations written in the form φ = ∑ a(α)φ(2 ·−α), α∈Z where the vector of functions φ =(φ1,...,φr) T is in (Lp(R)) r and a is a finitely supported sequence of r × r matrices called the refinement mask. Associated with the mask a is a linear o ..."
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Cited by 38 (13 self)
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Abstract. We consider solutions of a system of refinement equations written in the form φ = ∑ a(α)φ(2 ·−α), α∈Z where the vector of functions φ =(φ1,...,φr) T is in (Lp(R)) r and a is a finitely supported sequence of r × r matrices called the refinement mask. Associated with the mask a is a linear operator Qa defined on (Lp(R)) r by Qaf: = ∑ α∈Z a(α)f(2 ·−α). This paper is concerned with the convergence of the subdivision scheme associated with a, i.e., the convergence of the sequence (Qn a f)n=1,2,... in the Lpnorm. Our main result characterizes the convergence of a subdivision scheme associated with the mask a in terms of the joint spectral radius of two finite matrices derived from the mask. Along the way, properties of the joint spectral radius and its relation to the subdivision scheme are discussed. In particular, the L2convergence of the subdivision scheme is characterized in terms of the spectral radius of the transition operator restricted to a certain invariant subspace. We analyze convergence of the subdivision scheme explicitly for several interesting classes of vector refinement equations. Finally, the theory of vector subdivision schemes is used to characterize orthonormality of multiple refinable functions. This leads us to construct a class of continuous orthogonal double wavelets with symmetry. 1.