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