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Walsh Series, An Introduction to Dyadic Harmonic Analysis [Kõva köide]

  • Formaat: Hardback, 528 pages
  • Ilmumisaeg: 01-Oct-1990
  • Kirjastus: Institute of Physics Publishing
  • ISBN-10: 075030068X
  • ISBN-13: 9780750300681
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Walsh Series, An Introduction to Dyadic Harmonic AnalysisSuurem pilt
  • Formaat: Hardback, 528 pages
  • Ilmumisaeg: 01-Oct-1990
  • Kirjastus: Institute of Physics Publishing
  • ISBN-10: 075030068X
  • ISBN-13: 9780750300681
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9780750300681.html
Märksõnad:
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.
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.