Foundations of Discrete Harmonic Analysis, 1st ed. 2020 Lecture Notes in Applied and Numerical Harmonic Analysis Series

This book provides an introduction to discrete harmonic analysis (DHA) with a view towards applications to digital signal processing. In a nutshell, DHA is used to determine the time-frequency structure of a digitized signal, providing a representation of the signal as a sum of spectral components that can then be analyzed.

The main methods of DHA are discrete Fourier transform and other discrete orthogonal transforms such as the Walsh and Haar transforms. Fast algorithms are used to process signals in real time, while additional options are provided by spline harmonic analysis. These topics are carefully covered in the book. With only modest prerequisites, some of which are recalled at the beginning, a profound mathematical theory is built almost from scratch. The 150 exercises included form an integral part of the text.

Based decades of teaching experience, this book provides a basis for lecture courses starting at the upper undergraduate level, and will also prove a valuable resource for mathematicians and engineers interested in digital signal processing.

Vasily N. Malozemov, Doctor of Science, is professor at the Mathematics and Mechanics Faculty of Saint Petersburg University. His main scientific interests lie in the field of discrete harmonic analysis, spline theory, Chebyshev approximations and nonsmooth optimization, and fast algorithms.

Sergey M. Masharsky, Ph. D., is associate professor at the Mathematics and Mechanics Faculty of Saint Petersburg University. His main scientific interests lie in the field of discrete harmonic analysis and fast algorithms.

Provides a self-contained introduction to discrete harmonic analysis

Builds an advanced theory from scratch that can be applied to signal processing

Includes about 150 exercises, with solutions