Walsh Series and Transforms

1 Walsh Functions and Their Generalizations.- x1.1 The Walsh functions on the interval [0, 1).- x1.2 The Walsh system on the group.- x1.3 Other definitions of the Walsh system. Its connection with the Haar system.- x1.4 Walsh series. The Dirichlet kernel.- x1.5 Multiplicative systems and their continual analogues.- 2 Walsh-Fourier Series Basic Properties.- x2.1 Elementary properties of Walsh-Fourier series. Formulae for partial sums.- x2.2 The Lebesgue constants.- x2.3 Moduli of continuity of functions and uniform convergence of Walsh-Fourier series.- x2.4 Other tests for uniform convergence.- x2.5 The localization principle. Tests for convergence of a Walsh-Fourier series at a point.- x2.6 The Walsh system as a complete, closed system.- x2.7 Estimates of Walsh-Fourier coefficients. Absolute convergence of Walsh-Fourier series.- x2.8 Fourier series in multiplicative systems.- 3 General Walsh Series and Fourier-Stieltjes Series Questions on Uniqueness of Representation of Functions by Walsh Series.- x3.1 General Walsh series as a generalized Stieltjcs series.- x3.2 Uniqueness theorems for representation of functions by pointwise convergent Walsh series.- x3.3 A localization theorem for general Walsh series.- x3.4 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.- x4.1 Linear methods of summation. Regularity of the arithmetic means.- x4.2 The kernel for the method of arithmetic means for Walsh- Fourier series.- x4.3 Uniform (C, 1) summability of Walsh-Fourier series of continuous functions.- x4.4 (C, 1) summability of Fourier-Stieltjes series.- 5 Operators in the Theory of Walsh-Fourier Series.- x5.1 Some information from the theory of operators on spaces of measurable functions.- x5.2 The Hardy-Littlewood maximal operator corresponding to sequences of dyadic nets.- x5.3 Partial sums of Walsh-Fourier series as operators.- x5.4 Convergence of Walsh-Fourier series in Lp[0, 1).- 6 Generalized Multiplicative Transforms.- x6.1 Existence and properties of generalized multiplicative transforms.- x6.2 Representation of functions in L1(0, ?) by their multiplicative transforms.- x6.3 Representation of functions in Lp(0, ?), 1 < p ? 2, by their multiplicative transforms.- 7 Walsh Series with Monotone Decreasing Coefficient.- x7.1 Convergence and integrability.- x7.2 Series with quasiconvex coefficients.- x7.3 Fourier series of functions in Lp.- 8 Lacunary Subsystems of the Walsh System.- x8.1 The Rademacher system.- x8.2 Other lacunary subsystems.- x8.3 The Central Limit Theorem for lacunary Walsh series.- 9 Divergent Walsh-Fourier Series Almost Everywhere Convergence of Walsh-Fourier Series of L2 Functions.- x9.1 Everywhere divergent Walsh-Fourier series.- x9.2 Almost everywhere convergence of Walsh-Fourier series of L2[0, 1) functions.- 10 Approximations by Walsh and Haar Polynomials.- x10.1 Approximation in uniform norm.- x10.2 Approximation in the Lp norm.- x10.3 Connections between best approximations and integrability conditions.- x10.4 Connections between best approximations and integrability conditions (continued).- x10.5 Best approximations by means of multiplicative and step functions.- 11 Applications of Multiplicative Series and Transforms to Digital Information Processing.- x11.1 Discrete multiplicative transforms.- x11.2 Computation of the discrete multiplicative transform.- x11.3 Applications of discrete multiplicative transforms to information compression.- x11.4 Peculiarities of processing two-dimensional numerical problems with discrete multiplicative transforms.- x11.5 A description of classes of discrete transforms which allow fast algorithms.- 12 Other Applications of Multiplicative Functions and Transforms.- x12.1 Construction of digital filters based on multiplicative transforms.- x12.2 Multiplicative holographic transformations for image processing.- x12.3 Solutions to certain optimization problems.- Appendices.- Appendix 1 Abelian groups.- Appendix 2 Metric spaces. Metric groups.- Appendix 3 Measure spaces.- Appendix 4 Measurable functions. The Lebesgue integral.- Appendix 5 Normed linear spaces. Hilbert spaces.- Commentary.- References.