Walsh Series and Transforms - A. Efimov

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Résumé :
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 dig...