The book describes the theory and application of wavelets and framelets. Particularly, it presents an extensive study of the design of various types of filter banks. The book is part of Springer’s “Applied and Numerical Harmonic Analysis” series. It provides a wide range of theorems and proofs, and applies them to algorithms in the design of filter banks.

The first chapter, on discrete framelet transforms (DFrT), introduces one-level and multilevel discrete transforms. It investigates three properties of standard DFrT: perfect reconstruction, sparsity, and stability. Reconstruction refers to the standard flow--input to decomposition to reconstruction to output--in a one-level DFrT-based filter bank. The chapter includes techniques to achieve sparsity in smooth signals in DFrT; the stability conditions for multilevel DFrT; the oblique extension principle; and computable formulas for bounded signals.

The second chapter is on the design of wavelet filter banks. The first few sections of the chapter discuss properties such as interpolation, symmetry, linear-phase moments, and minimal supports. Later sections present algorithms for orthogonal and biorthogonal wavelet filter banks; proofs of correctness, examples, and variants of the algorithms with respect to filter bank properties are included.

The third chapter discusses the design of framelet filter banks. It presents dual framelet filter banks and tight framelet filter banks. The designs are based on Laurent polynomials. It describes algorithms for designs with properties such as symmetry, phase moments, and interpolation.

The next three chapters present framelet/wavelet theories. They describe framelets/wavelets with respect to affine systems, Hilbert spaces, shift invariant subspaces, Sobolev spaces, periods, refinable functions, and so on. They explain the connections between various types of filter banks and discuss the convergence of cascade algorithms in filter banks using these theories.

The last chapter is on other applications of wavelets/framelets, including “subdivision schemes in computer graphics,” image processing, finite intervals for numerical algorithms, and so on. The chapter begins with multidimensional framelets/wavelets. An appendix at the end gives the fundamentals of Fourier analysis.

Readers must have an understanding of signal processing, filters, and filter banks. On the mathematics side, knowledge of basic statistics, Fourier analysis, and functions is required. On the computing side, a background in design and algorithms is necessary. The author describes in the preface how the chapters can be used in various types of courses for graduate students or junior researchers.