Characterization of Matrices Generating Orthogonal Multi-wavelets
ABSTRACT Theoretically, multiwavelets hold significant advantages over standard wavelets, particularly for solving more complicated problems, and hence are of great interest. It is pointed out in this paper that we must avoid those "multi-wavelets" being just pairs of standard wavelets. The main purpose of this paper is to charaterize a class of matrices which is crucial for the construction of orthogonal multi-wavelets with uniform linear phase.
- SourceAvailable from: Carl R de Boor[Show abstract] [Hide abstract]
ABSTRACT: A second look at the authors' [BDR1], [BDR2] characterization of the approximation order of a Finitely generated Shift-Invariant (FSI) subspace of L 2(R d ) 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 contains a (necessarily unique) satisfying for |j|<k , . The technical condition is satisfied, e.g., when the generators are at infinity for some >k+d. In the case of compactly supported generators, this recovers an earlier result of Jia [J1], [J2].Constructive Approximation 01/1998; 14(4):631-652. · 1.07 Impact Factor
- SIAM J. Matrix Analysis Applications. 01/2003; 25:517-531.
Book: Matrix Analysis[Show abstract] [Hide abstract]
ABSTRACT: The book under review is devoted to matrix analysis in the spirit of functional analysis and with great emphasis on the art of deriving matrix inequalities. This art can be compared to that of cutting diamonds: it requires hard tools and a delicate use of them. The text consists of ten chapters: The first chapter establishes notations and introduces preliminaries from linear and multilinear algebra. Special attention is given to tensor products and symmetry classes. In the next three chapters well elaborated background material is explained which should be included in any course on matrix analysis. Chapter 5 on operator monotone and operator convex functions presents more advanced and more special material. Chapters 6 and 8 are devoted to perturbation of spectra, a topic of much importance in numerical analysis, physics and engineering. Chapter 9 (selection of matrix inequalities) and Chapter 10 (Perturbation of matrix functions) also have been of broad interest in several areas. The presentation of the material covered in the book is very clear. The contents of each chapter are briefly summarized in its first paragraph, while at the end of each chapter detailed references and comments to the original published sources and the most important related papers are collected. These Notes and References give insight, explain the ideas behind many of the concepts, and point out connections. Several exercises are scattered in the text and in the Problem section in each chapter. These exercises range in difficulty from the “quite easy” to hard enough to yield the contents of research papers. On the whole, the author has managed to create a highly readable and attractive account of the subject. The book is a must for anyone working in matrix analysis; it can be recommended to graduate students as well as to specialists.01/1996; Springer.