This book provides a broadly based, theoretical monograph on the Walsh System, a system that is the simplest non-trivial model for harmonic analysis and shares many properties with the trigonometric system.
It gives a thorough introduction to foundations of Walsh-Fourier analysis introducing the main techniques and fundamental problems in a way that makes the literature accessible. It also shows how the theory of Walsh-Fourier analysis relates to other aspects of harmonic analysis.
The book will be of interest to postgraduate students in pure and applied mathematics, and those studying numerical analysis and computational mathematics.
Sprache
Verlagsort
Verlagsgruppe
Zielgruppe
Für höhere Schule und Studium
Für Beruf und Forschung
postgraduate students in pure and applied mathematics; those studying numerical analysis and computational mathematics; useful reference volume for anyone interested in the field.
ISBN-13
978-0-7503-0068-1 (9780750300681)
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Schweitzer Klassifikation
Provisional. The Walsh Functions. The dyadic group. The representation of the dyadic group on the interval (0,1). Transformations and rearrangements of the Walsh system. Walsh-Fourier coefficients. Walsh-Fourier series and the Dirichlet kernel. The dyadic derivative. Summability. Exercises. Walsh-Fourier Coefficients: estimates of Walsh-Fourier coefficients. Walsh-Fourier coefficients of absolutely continuous functions. Walsh-Fourier coefficients of continuous functions. Absolute convergence of Walsh-Fourier series. Pointwise convergence and summability. Exercises. Dyadic martingales and Hardy Spaces: Dyadic martingales and the dyadic maximal function. An interpolation theorem and the canonical decomposition. Martingale transforms. Dyadic Hardy spaces and BMO. Quality in dyadic Hardy spaces. Martingale trees and a.e. convergence of Walsh-Fourier series. Exercises. Convergence in norm: P convergence of Walsh-Fourier series. Uniform convergence of Walsh-Fourier series. The Walsh-Fejer polynomials. Summability of Walsh-Fourier series in homogeneous Banach spaces. Sets of divergence. Adjustment of functions. Exercises. Approximation and Bases: approximation by Walsh polynomials. The strong derivative and approximation. The Haar, Walsh, and Faber-Schauder systems as bases. The Franklin system. Equivalence of bases. The basis problem. Exercises. A.e. convergence and summability of Walsh-Fourier series: tests for a.e. convergence. A.e summability of Walsh-Fourier series and pointwise dyadic derivative. Logarithm spaces and block spaces. A.e. convergence of rearrangement of Walsh-Fourier series and closely related systmes. Divergent Walsh-Fourier series. A.e. convergence of double Walsh-Fourier series. Exercises. Uniqueness: Walsh series and quasi-measures. Uniqueness of a.e. convergent Walsh series. Null series and the formal product. Null series and measure preserving transformations. U-sets and M-sets. Uniqueness and Cesaro summmability. Exercises. Respresentation by Walsh series: Walsh series with monotone coefficients. Term by term dyadic differentiation. Representation by measurable functions. Normalized convergence systems. The Walsh-Fourier Transofrm. The dyadic field. The Walsh-Fourier transform. The Walsh-Fourier-Plancherel transform. Inversion of the Walsh-Fourier transform. The inverse dyadic derivative. The Mellin transform. The fast Walsh transfrom. Exercises. Appendices.
