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 ofmeasurable 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 < p ? 2, by their multiplicative transforms.- 7 Walsh Series with Monotone Decreasing Coefficient.- §7.1 Convergence and integrability.- §7.2 Series with quasiconvex coefficients.- §7.3 Fourier series of functions in Lp.- 8 Lacunary Subsystems of the Walsh System.- §8.1 The Rademacher system.- §8.2 Other lacunary subsystems.- §8.3 The Central Limit Theorem for lacunary Walsh series.- 9 Divergent Walsh-Fourier Series Almost Everywhere Convergence of Walsh-Fourier Series of L2 Functions.- §9.1 Everywhere divergent Walsh-Fourier series.- §9.2 Almost everywhere convergence of Walsh-Fourier series of L2[0, 1) functions.- 10 Approximations by Walsh and Haar Polynomials.- §10.1 Approximation in uniform norm.- §10.2 Approximation in the Lp norm.- §10.3 Connections between best approximations and integrability conditions.- §10.4 Connections between best approximations and integrability conditions (continued).- §10.5 Best approximations by means of multiplicative and step functions.- 11 Applications of Multiplicative Series and Transforms to Digital Information Processing.- §11.1 Discrete multiplicative transforms.- §11.2 Computation of the discrete multiplicative transform.- §11.3 Applications of discrete multiplicative transforms to information compression.- §11.4 Peculiarities ofprocessing two-dimensional numerical problems with discrete multiplicative transforms.- §11.5 A description of classes of discrete transforms which allow fast algorithms.- 12 Other Applications of Multiplicative Functions and Transforms.- §12.1 Construction of digital filters based on multiplicative transforms.- §12.2 Multiplicative holographic transformations for image processing.- §12.3 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.