Walsh Series and Transforms: Theory and Applications [Paperback]

1 Walsh Functions and Their Generalizations.- ?1.1 The Walsh functions on the interval [0, 1).- ?1.2 The Walsh system on the group.- ?1.3 Other definitions of the Walsh system. Its connection with the Haar system.- ?1.4 Walsh series. The Dirichlet kernel.- ?1.5 Multiplicative systems and their continual analogues.- 2 Walsh-Fourier Series Basic Properties.- ?2.1 Elementary properties of Walsh-Fourier series. Formulae for partial sums.- ?2.2 The Lebesgue constants.- ?2.3 Moduli of continuity of functions and uniform convergence of Walsh-Fourier series.- ?2.4 Other tests for uniform convergence.- ?2.5 The localization principle. Tests for convergence of a Walsh-Fourier series at a point.- ?2.6 The Walsh system as a complete, closed system.- ?2.7 Estimates of Walsh-Fourier coefficients. Absolute convergence of Walsh-Fourier series.- ?2.8 Fourier series in multiplicative systems.- 3 General Walsh Series and Fourier-Stieltjes Series Questions on Uniqueness of Representation of Functions by Walsh Series.- ?3.1 General Walsh series as a generalized Stieltjcs series.- ?3.2 Uniqueness theorems for representation of functions by pointwise convergent Walsh series.- ?3.3 A localization theorem for general Walsh series.- ?3.4 Examples of null series in the Walsh system. The concept of U-sets and M-sets.- 4 Summation of Walsh Series by the Method of Arithmetic Mean.- ?4.1 Linear methods of summation. Regularity of the arithmetic means.- ?4.2 The kernel for the method of arithmetic means for Walsh- Fourier series.- ?4.3 Uniform (C, 1) summability of Walsh-Fourier series of continuous functions.- ?4.4 (C, 1) summability of Fourier-Stieltjes series.- 5 Operators in the Theory of Walsh-Fourier Series.- ?5.1 Some information from the theory of operators on spaces of measurable functions.- ?5.2 The Hardy-Littlewood maximal operator corresponding to sequences of dyadic nets.- ?5.3 Partial sums of Walsh-Fourier series as operators.- ?5.4 Convergence of Walsh-Fourier series in Lp[0, 1).- 6lC%