Walsh series and transforms : theory and applications
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Bibliographic Information
Walsh series and transforms : theory and applications
（Mathematics and its applications, . Soviet series ; v. 64）
Kluwer Academic Publishers, c1991
- Other Title
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Ряды и преобразования уолша : теория и примeнения
Ri︠a︡dy i preobrazovanii︠a︡ Uolsha
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Translation of: Ri︠a︡dy i preobrazovanii︠a︡ Uolsha
Includes bibliographical references (p. 359-364) and index
Description and Table of Contents
Description
'Et moi, ..., si j'avait su comment en revenir, One service mathematics has rendered the je n'y se.rais point aile.' human race. It has put common sense back Jules Verne where it belongs, on !be topmost shelf next to the dusty canister labelled 'disc:arded non- sense'. The series is divergent; therefore we may be able to do something with it. Eric T. Bell O. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non- linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics ...'; 'One service logic has rendered com- puter science ...'; 'One service category theory has rendered mathematics ...'. All arguably true. And all statements obtainable this way form part of the raison d'etre of this series.
Table of Contents
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 of measurable 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 of processing 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.
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