Framelets and wavelets. Algorithms, analysis, and applications
This monograph provides a comprehensive introduction to the mathematical theory of framelets and discrete framelet transforms. In the classical wavelet theory, a \textit{wavelet} is a function set \(\Psi \subset L_2(\mathbb R)\), where the related affine system \[ \{\psi_{j,k}(x) = 2^{j/2}\,\psi(2^j x - k) :\, j,k\in \mathbb Z\,, \,\psi \in \Psi\} \] is a basis of \(L_2(\mathbb R)\). Now two function sets \(\Phi\), \(\Psi \subset L_2(\mathbb R)\) form a \textit{framelet}, if for \(J\in \mathbb Z\) the corresponding affine system \[ \{\varphi_{J,k} :\,k\in \mathbb Z\,,\,\varphi\in \Phi\} \cap\{\psi_{j,k} :\, k\in \mathbb Z\,,j \geq J\,, \, \psi \in \Psi\} \] is a frame of \(L_2(\mathbb R)\), where a frame is an ``overcomplete basis''. Thus a wavelet is merely a framelet without redundancy. The author has made important contributions to framelet theory. His book systematically introduces the theory of framelets from the perspective of discrete framelet transforms such that multiresolution analysis plays only a marginal role in this theory. This monograph consists of seven chapters and an appendix. Chapter 1 describes the properties of discrete framelet/wavelet transforms and the corresponding implementation. Chapter 2 presents algorithms for constructing wavelet filter banks with/without symmetry. Chapter 3 develops general algorithms for constructing all possible dual framelet filter banks and tight framelet filter banks with/without symmetry such that the high-pass filters have short filter supports. In Chapter 4, the author investigates affine systems and dual framelets in the univariate case. In Chapter 5, he studies special refinable vector functions and their relations to affine systems derived from refinable functions. Multivariate scalar framelets/wavelets are studied in Chapter 7, where subdivision schemes and their applications to curve/surface generation are discussed. The Appendix presents some basic results on function spaces and Fourier analysis. The theory is illustrated by numerous examples and figures. More than 360 exercises are formulated. Historical comments are given to each chapter. This monograph is well-written for a broad readership and very convenient as a textbook for graduate students and as an advanced reference guide for researchers in applied mathematics, physics, and engineering. Doubtless, this work will stimulate further research on framelets.
scientific article; zbMATH DE number 6798658