More than 80 years ago, J. L. Walsh introduced and studied a certain orthogonal system of step functions which are now widely used in connection with so-called Walsh functions and Fourier-Walsh transforms. To the reviewer’s best knowledge, the textbook under review is the first systematic and self-contained account of the theory, methods, and applications of Walsh functions and their various generalizations.
The book consists of 6 chapters with the following headings: 1. The Walsh transform; 2. Dyadic integrals and derivatives on the half-line; 3. Dyadic integrals and derivatives of fractional order; 4. A dyadic analogue to Wiener’s Tauberian theorem; 5. Dyadic Hardy and Hardy-Littlewood operators; 6. Dyadic distributions.
This textbook will be of interest to all specialists in functional analysis, operator theory, harmonic analysis, Fourier analysis, and approximation theory. A translation into English would perhaps provide it with the larger readership it deserves.