Abstract not found

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 shift-invariant approximations such as those provided by splines and wavelets, as well as finite elements and multi-wavelets 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 multi-wavelet space and the order of a particular subspace generated by a single function. Finally, we consider refinable generating functions and use the two-scale 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 multi-wavelet expansions of a given order L.

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 Lp-norm. 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 L2-convergence 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.

. 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...

. 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 two-scale 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 ...

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.

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.

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 consider the existence of distributional (or L 2 ) solutions of the matrix refinement equation b \Phi = P(\Delta=2) b \Phi(\Delta=2); where P is an r \Theta r matrix with trigonometric polynomial entries. One of the main results of this paper is that the above matrix refinement equation has a compactly supported distributional solution if and only if the matrix P(0) has an eigenvalue of the form 2 n , n 2 ZZ + . A characterization of the existence of L 2 -solutions of the above matrix refinement equation in terms of the mask is also given. A concept of L 2 -weak stability of a (finite) sequence of function vectors is introduced. In the case when the function vectors are solutions of a matrix refinement equation, we characterize this weak stability in terms of the mask. AMS Subject Classification: Primary 42C15, 42B05, 41A30 Secondary 39B62, 42B10 Keywords: Refinable function vectors, stable basis y Permanent address: Department of Mathematics, Peking University. 1. Introducti...

In a previous work [32] we investigated how to reverse subdivision rules using global least squares fitting. This led to multiresolution structures that could be viewed as semiorthogonal wavelet systems whose inner product was that for finite-dimensional Cartesian vector space. We produced simple and sparse reconstruction filters, but had to appeal to matrix factorization to obtain an efficient, exact decomposition. We also made some observations on how the inner product that defines the semiorthogonality inuences the sparsity of the reconstruction filters. In this work we carry the investigation further by studying biorthogonal systems based upon subdivision rules and local least squares fitting problems that reverse the subdivision. We are able to produce multiresolution structures for some common univariate subdivision rules that have both sparse reconstruction and decomposition filters. Three will be presented here -- for quadratic and cubic B-spline subdivision and for the 4-point interpola...