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# Walsh Series and Transforms: Theory and Applications

## byB. Golubov, A. Efimov, V. Skvortsov

### Paperback | October 30, 2012

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'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.

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Title:Walsh Series and Transforms: Theory and ApplicationsFormat:PaperbackDimensions:368 pagesPublished:October 30, 2012Publisher:Springer-Verlag/Sci-Tech/TradeLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:9401054525

ISBN - 13:9789401054522

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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 <_20_p20_3f_20_22c_20_by20_their20_multiplicative20_transforms.-20_720_walsh20_series20_with20_monotone20_decreasing20_coefficient.-20_c2a7_7.120_convergence20_and20_integrability.-20_c2a7_7.220_series20_with20_quasiconvex20_coefficients.-20_c2a7_7.320_fourier20_series20_of20_functions20_in20_lp.-20_820_lacunary20_subsystems20_of20_the20_walsh20_system.-20_c2a7_8.120_the20_rademacher20_system.-20_c2a7_8.220_other20_lacunary20_subsystems.-20_c2a7_8.320_the20_central20_limit20_theorem20_for20_lacunary20_walsh20_series.-20_920_divergent20_walsh-fourier20_series20_almost20_everywhere20_convergence20_of20_walsh-fourier20_series20_of20_l220_functions.-20_c2a7_9.120_everywhere20_divergent20_walsh-fourier20_series.-20_c2a7_9.220_almost20_everywhere20_convergence20_of20_walsh-fourier20_series20_of20_l25b_02c_20_129_20_functions.-20_1020_approximations20_by20_walsh20_and20_haar20_polynomials.-20_c2a7_10.120_approximation20_in20_uniform20_norm.-20_c2a7_10.220_approximation20_in20_the20_lp20_norm.-20_c2a7_10.320_connections20_between20_best20_approximations20_and20_integrability20_conditions.-20_c2a7_10.420_connections20_between20_best20_approximations20_and20_integrability20_conditions20_28_continued29_.-20_c2a7_10.520_best20_approximations20_by20_means20_of20_multiplicative20_and20_step20_functions.-20_1120_applications20_of20_multiplicative20_series20_and20_transforms20_to20_digital20_information20_processing.-20_c2a7_11.120_discrete20_multiplicative20_transforms.-20_c2a7_11.220_computation20_of20_the20_discrete20_multiplicative20_transform.-20_c2a7_11.320_applications20_of20_discrete20_multiplicative20_transforms20_to20_information20_compression.-20_c2a7_11.420_peculiarities20_of20_processing20_two-dimensional20_numerical20_problems20_with20_discrete20_multiplicative20_transforms.-20_c2a7_11.520_a20_description20_of20_classes20_of20_discrete20_transforms20_which20_allow20_fast20_algorithms.-20_1220_other20_applications20_of20_multiplicative20_functions20_and20_transforms.-20_c2a7_12.120_construction20_of20_digital20_filters20_based20_on20_multiplicative20_transforms.-20_c2a7_12.220_multiplicative20_holographic20_transformations20_for20_image20_processing.-20_c2a7_12.320_solutions20_to20_certain20_optimization20_problems.-20_appendices.-20_appendix20_120_abelian20_groups.-20_appendix20_220_metric20_spaces.20_metric20_groups.-20_appendix20_320_measure20_spaces.-20_appendix20_420_measurable20_functions.20_the20_lebesgue20_integral.-20_appendix20_520_normed20_linear20_spaces.20_hilbert20_spaces.-20_commentary.-20_references. p="" _22c_="" by="" their="" multiplicative="" transforms.-="" 7="" walsh="" series="" with="" monotone="" decreasing="" coefficient.-="" _c2a7_7.1="" convergence="" and="" integrability.-="" _c2a7_7.2="" quasiconvex="" coefficients.-="" _c2a7_7.3="" fourier="" of="" functions="" in="" lp.-="" 8="" lacunary="" subsystems="" the="" system.-="" _c2a7_8.1="" rademacher="" _c2a7_8.2="" other="" subsystems.-="" _c2a7_8.3="" central="" limit="" theorem="" for="" series.-="" 9="" divergent="" walsh-fourier="" almost="" everywhere="" l2="" functions.-="" _c2a7_9.1="" _c2a7_9.2="" _l25b_02c_="" _129_="" 10="" approximations="" haar="" polynomials.-="" _c2a7_10.1="" approximation="" uniform="" norm.-="" _c2a7_10.2="" lp="" _c2a7_10.3="" connections="" between="" best="" integrability="" conditions.-="" _c2a7_10.4="" conditions="" _28_continued29_.-="" _c2a7_10.5="" means="" step="" 11="" applications="" transforms="" to="" digital="" information="" processing.-="" _c2a7_11.1="" discrete="" _c2a7_11.2="" computation="" transform.-="" _c2a7_11.3="" compression.-="" _c2a7_11.4="" peculiarities="" processing="" two-dimensional="" numerical="" problems="" _c2a7_11.5="" a="" description="" classes="" which="" allow="" fast="" algorithms.-="" 12="" _c2a7_12.1="" construction="" filters="" based="" on="" _c2a7_12.2="" holographic="" transformations="" image="" _c2a7_12.3="" solutions="" certain="" optimization="" problems.-="" appendices.-="" appendix="" 1="" abelian="" groups.-="" 2="" metric="" spaces.="" 3="" measure="" spaces.-="" 4="" measurable="" functions.="" lebesgue="" integral.-="" 5="" normed="" linear="" hilbert="" commentary.-="">