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60
Approximation error for quasiinterpolators and (multi)wavelet expansions
 APPL. COMPUT. HARMON. ANAL
, 1999
"... We investigate the approximation properties of general polynomial preserving operators that approximate a function into some scaled subspace of L² via an appropriate sequence of inner products. In particular, we consider integer shiftinvariant approximations such as those provided by splines and wa ..."
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Cited by 64 (22 self)
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We investigate the approximation properties of general polynomial preserving operators that approximate a function into some scaled subspace of L² via an appropriate sequence of inner products. In particular, we consider integer shiftinvariant approximations such as those provided by splines and wavelets, as well as finite elements and multiwavelets which use multiple generators. We estimate the approximation error as a function of the scale parameter T when the function to approximate is sufficiently regular. We then present a generalized sampling theorem, a result that is rich enough to provide tight bounds as well as asymptotic expansions of the approximation error as a function of the sampling step T. Another more theoretical consequence is the proof of a conjecture by Strang and Fix, which states the equivalence between the order of a multiwavelet space and the order of a particular subspace generated by a single function. Finally, we consider refinable generating functions and use the twoscale relation to obtain explicit formulae for the coefficients of the asymptotic development of the error. The leading constants are easily computable and can be the basis for the comparison of the approximation power of wavelet and multiwavelet expansions of a given order L.
Vector cascade algorithms and refinable function vectors in Sobolev spaces
 J. Approx. Theory
, 2002
"... In this paper we shall study vector cascade algorithms and refinable function vectors with a general isotropic dilation matrix in Sobolev spaces. By investigating several properties of the initial function vectors in a vector cascade algorithm, we are able to take a relatively unified approach to st ..."
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Cited by 54 (35 self)
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In this paper we shall study vector cascade algorithms and refinable function vectors with a general isotropic dilation matrix in Sobolev spaces. By investigating several properties of the initial function vectors in a vector cascade algorithm, we are able to take a relatively unified approach to study several questions such as convergence, rate of convergence and error estimate for a perturbed mask of a vector cascade algorithm in a Sobolev space W k p (R s)(1 � p � ∞, k ∈ N∪{0}). We shall characterize the convergence of a vector cascade algorithm in a Sobolev space in various ways. As a consequence, a simple characterization for refinable Hermite interpolants and a sharp error estimate for a perturbed mask of a vector cascade algorithm in a Sobolev space will be presented. The approach in this paper enables us to answer some unsolved questions in the literature on vector cascade algorithms and to comprehensively generalize and improve results on scalar cascade algorithms and scalar refinable functions to the vector case. Key words: vector cascade algorithm, vector subdivision scheme, refinable function vector, Hermite interpolant, initial function vector, error estimate, sum rules, smoothness.
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 43 (14 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.
Spectral analysis of the transition operator and its applications to smoothness analysis of wavelets
 SIAM J. Matrix. Anal. Appl
, 2001
"... The purpose of this paper is to investigate spectral properties of the transition operator associated to a multivariate vector refinement equation and their applications to the study of smoothness of the corresponding refinable vector of functions. Let Φ = (φ1,..., φr) T be an r × 1 vector of compac ..."
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Cited by 43 (17 self)
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The purpose of this paper is to investigate spectral properties of the transition operator associated to a multivariate vector refinement equation and their applications to the study of smoothness of the corresponding refinable vector of functions. Let Φ = (φ1,..., φr) T be an r × 1 vector of compactly supported functions in L2(IR s) satisfying the refinement equation Φ = � α ∈ Zs a(α)Φ(M · − α), where M is an expansive integer matrix. We assume that M is isotropic, i.e., M is similar to a diagonal matrix diag(σ1,..., σs) with σ1  = · · · = σs. For µ = (µ1,..., µs) ∈ IN s 0, define. The smoothness of Φ is measured by the critical exponent σ −µ: = σ −µ1
Analysis And Construction Of Optimal Multivariate Biorthogonal Wavelets With Compact Support
 SIAM J. Math. Anal
, 1998
"... . In applications, it is well known that high smoothness, small support and high vanishing moments are the three most important properties of a biorthogonal wavelet. In this paper, we shall investigate the mutual relations among these three properties. A characterization of Lp (1 p 1) smoothness ..."
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Cited by 43 (33 self)
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. In applications, it is well known that high smoothness, small support and high vanishing moments are the three most important properties of a biorthogonal wavelet. In this paper, we shall investigate the mutual relations among these three properties. A characterization of Lp (1 p 1) smoothness of multivariate refinable functions is presented. It is well known that there is a close relation between a fundamental refinable function and a biorthogonal wavelet. We shall demonstrate that any fundamental refinable function, whose mask is supported on [1 \Gamma 2r; 2r \Gamma 1] s for some positive integer r and satisfies the sum rules of optimal order 2r, has Lp smoothness not exceeding that of the univariate fundamental refinable function with the mask br . Here the sequence br on Z is the unique univariate interpolatory refinement mask which is supported on [1 \Gamma 2r; 2r \Gamma 1] and satisfies the sum rules of order 2r. Based on a similar idea, we shall prove that any orthogonal...
Construction of Multiscaling Functions with Approximation and Symmetry
, 1998
"... . This paper presents a new and e#cient way to create multiscaling functions with given approximation order, regularity, symmetry, and short support. Previous techniques were operating in time domain and required the solution of large systems of nonlinear equations. By switching to the frequency dom ..."
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Cited by 42 (10 self)
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. This paper presents a new and e#cient way to create multiscaling functions with given approximation order, regularity, symmetry, and short support. Previous techniques were operating in time domain and required the solution of large systems of nonlinear equations. By switching to the frequency domain and employing the latest results of the multiwavelet theory we are able to elaborate a simple and e#cient method of construction of multiscaling functions. Our algorithm is based on a recently found factorization of the refinement mask through the twoscale similarity transform (TST). Theoretical results and new examples are presented. Key words. approximation order, symmetry, multiscaling functions, multiwavelets AMS subject classifications. 41A25, 42A38, 39B62 PII. S0036141096297182 1. Introduction. This paper discusses the construction of multiscaling functions which generate a multiresolution analysis (MRA) and lead to multiwavelets. A standard (scalar) MRA assumes that there is ...
Multiscale Decompositions on Bounded Domains
 TRANS. AMER. MATH. SOC
, 1995
"... A construction of multiscale decompositions relative to domains\Omega ae IR d is given. Multiscale spaces are constructed on\Omega which retain the important features of univariate multiresolution analysis including local polynomial reproduction and locally supported, stable bases. ..."
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Cited by 41 (12 self)
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A construction of multiscale decompositions relative to domains\Omega ae IR d is given. Multiscale spaces are constructed on\Omega which retain the important features of univariate multiresolution analysis including local polynomial reproduction and locally supported, stable bases.
Approximation Properties and Construction of Hermite Interpolants and Biorthogonal Multiwavelets
, 2001
"... ..."
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 35 (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 ...