Construction of wavelets through Walsh functions
This book focuses on the fertile interplay between Walsh functions and wavelet analysis. It contains nine chapters and an appendix. Chapter 1 discusses the basic results of Walsh analysis and wavelets on the real line. Chapter 2 deals with the growth and decay of Walsh-Fourier coefficients for classes of functions, convergence and summability of Walsh-Fourier series and the rate of approximation for different classes of functions and different means of Walsh-Fourier series, and applications to signal and image processing. Chapter 3 presents the Haar system and its generalization the Haar-Fourier series, the relation between the Haar and Walsh functions, nonuniform Haar wavelets, generalized Haar wavelets and frames, as well as applications of Haar wavelets to the solution of initial value problems, boundary value problems and integral equations. Chapter 4 deals with the construction of dyadic wavelets and frames through Walsh functions. The first section covers basic results needed in subsequent chapters, the second section explains orthogonal wavelets and multiresolution analysis on the half line, the third section covers orthogonal wavelets with compact support on the half line, the fourth section presents estimates for the smoothness of the scaling functions, and the fifth and last section presents approximation properties of dyadic wavelets. Chapter 5 is devoted to orthogonal and periodic wavelets on Vilenkin groups. The first section deals with multiresolution analysis on Vilenkin groups, section 2 introduces compactly supported orthogonal $p$-wavelets, section 3 presents periodic wavelets on Vilenkin groups, and section 4 introduces periodic wavelets related to the Vilenkin-Christenson transform; finally, section 5 discusses applications to the coding of fractal functions. Chapter 6 studies Haar-Vilenkin wavelets and their properties. Its five sections are titled as follows: Introduction, Haar-Vilenkin wavelets, Approximation by Haar-Vilenkin wavelets, Convergence theorems, and Haar-Vilenkin coefficients. Chapter 7 studies basic properties of biorthogonal wavelets and frames on the Vilenkin group, and applications to image processing. Chapter 8 studies wavelets associated with nonuniform multiresolution analyses on the half line, in particular the construction of a nonuniform multiresolution analysis and the analogue of Cohen's condition, as well as nonuniform wavelet frames in $L^2(R)$. The title of Chapter 9 is ``Orthogonal vector-valued wavelets on $R_+$'', but it begins by studying vector-valued multiresolution analysis on $R$. It then studies vector-valued multiresolution $p$-analysis on $R_+$, the existence of orthogonal vector-valued wavelets on $R_+$, vector-valued nonuniform multiresolution analysis, vector-valued nonuniform wavelet packets and their properties, and vector valued nonuniform wavelet bases. The appendix contains some of the basic results of functional analysis, Vilenkin systems, the Vilenkin-Pontryagin class of functions, and real and Fourier analysis. \par This book has an extensive bibliography of nearly 200 items and will be a valuable reference source. Since most chapters have exercises it could also be used as a textbook in an advanced graduate course. This work is noteworthy for its thoroughness, clarity, and historical perspective.
orthogonal and periodic wavelets on Vilenkin groups
vector-valued nonuniform wavelet packets and bases
orthogonal wavelets and multiresolution analysis on the half line
orthogonal vector-valued wavelets on the half line
scientific article; zbMATH DE number 7024146