Walsh Series and Transforms

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).- 6 Generalized Multiplicative Transforms.- §6.1 Existence and properties of generalized multiplicative transforms.- §6.2 Representation of functions in L1(0, ?) by their multiplicative transforms.- §6.3 Representation of functions in Lp(0, ?), 1