Walsh Series and Transforms

Inhaltsverzeichnis

• 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.