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Convergence and Summability of Fourier Transforms and Hardy Spaces

This book investigates the convergence and summability of both one-dimensional and multi-dimensional Fourier transforms, as well as the theory of Hardy spaces. To do so, it studies a general summability method known as theta-summation, which encompasses all the well-known summability methods, such a...

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Bibliographic Details
Main Author: Weisz, Ferenc
Format: eBook
Language:English
Published: Cham Springer International Publishing 2017, 2017
Edition:1st ed. 2017
Series:Applied and Numerical Harmonic Analysis
Subjects:
Fourier Analysis
Harmonic Analysis
Sequences, Series, Summability
Abstract Harmonic Analysis
Sequences (mathematics)
Online Access:
Collection: Springer eBooks 2005- - Collection details see MPG.ReNa
Table of Contents:
  • List of Figures
  • Preface
  • I One-dimensional Hardy spaces and Fourier transforms
  • 1 One-dimensional Hardy spaces
  • 1.1 The Lp spaces
  • 1.2 Hardy-Littlewood maximal function
  • 1.3 Schwartz functions
  • 1.4 Tempered distributions and Hardy spaces
  • 1.5 Inequalities with respect to Hardy spaces
  • 1.6 Atomic decomposition
  • 1.7 Interpolation between Hardy spaces
  • 1.8 Bounded operators on Hardy spaces
  • 2 One-dimensional Fourier transforms
  • 2.1 Fourier transforms
  • 2.2 Tempered distributions
  • 2.3 Partial sums of Fourier series
  • 2.4 Convergence of the inverse Fourier transform
  • 2.5 Summability of one-dimensional Fourier transforms
  • 2.6 Norm convergence of the summability means
  • 2.7 Almost everywhere convergence of the summability means
  • 2.8 Boundedness of the maximal operator
  • 2.9 Convergence at Lebesgue points
  • 2.10 Strong summability
  • 2.11 Some summability methods
  • II Multi-dimensional Hardy spaces and Fourier transforms
  • 5 `q-summability of multi-dimensional Fourier transforms
  • 5.1 The `-summability means
  • 5.2 Norm convergence of the `q-summability means
  • 5.2.1 Proof ofTheorem 5.2.1 for q = 1 and q = 1
  • 5.2.1.1 Proof for q = 1 in the two-dimensional case
  • 5.2.1.2 Proof for q = 1 in higher dimensions (d 3)
  • 5.2.1.3 Proof for q = 1 in the two-dimensional case
  • 5.2.1.4 Proof for q = 1 in higher dimensions (d 3)
  • 5.2.2 Some summability methods
  • 5.2.3 Further results for the Bochner-Riesz means
  • 5.3 Almost everywhere convergence of the `q-summability means
  • 5.3.1 Proof of Theorem 5.3.2
  • 5.3.1.1 Proof for q = 1 in the two-dimensional case
  • 5.3.1.2 Proof for q = 1 in higher dimensions (d 3)
  • 5.3.1.3 Proof for q = 1 in the two-dimensional case
  • 5.3.1.4 Proof for q = 1 in higher dimensions (d 3)
  • 5.3.2 Proof of Theorem 5.3.3
  • 5.3.3 Some summability methods
  • 5.3.4 Further results for the Bochner-Riesz means
  • 5.4 Convergence at Lebesgue points
  • 3 Multi-dimensional Hardy spaces
  • 3.1 Multi-dimensional maximal functions
  • 3.1.1 Hardy-Littlewood maximal functions
  • 3.1.2 Strong maximal functions
  • 3.2 Multi-dimensional tempered distributions and Hardy spaces
  • 3.3 Inequalities with respect to multi-dimensional Hardy spaces
  • 3.4 Atomic decompositions
  • 3.4.1 Atomic decomposition of H2p (Rd)
  • 3.4.2 Atomic decomposition of Hp(Rd)
  • 3.5 Interpolation between multi-dimensional Hardy spaces
  • 3.5.1 Interpolation between the H2p (Rd) spaces
  • 3.5.2 Interpolation between the Hp(Rd) spaces
  • 3.6 Bounded operators on multi-dimensional Hardy spaces
  • 3.6.1 Bounded operators on H2p (Rd)
  • 3.6.2 Bounded operators on Hp(Rd)
  • 4 Multi-dimensional Fourier transforms
  • 4.1 Fourier transforms
  • 4.2 Multi-dimensional partial sums
  • 4.3 Convergence of the inverse Fourier transform
  • 4.4 Multi-dimensional Dirichlet kernels
  • 4.4.1 Triangular Dirichlet kernels
  • 4.4.2 Circular Dirichlet kernels
  • 5.4.1 Circular summability (q = 2)
  • 5.4.2 Cubic and triangular summability (q = 1 and q = 1)
  • 5.4.2.1 Proof of the results for q = 1 and d = 2
  • 5.4.2.2 Proof of the results for q = 1 and d = 2
  • 5.4.2.3 Proof of the results for q = 1 and d 3
  • 5.4.2.4 Proof of the results for q = 1 and d 3
  • 5.5 Proofs of the one-dimensional strong summability results
  • 6 Rectangular summability of multi-dimensional Fourier transforms
  • 6.1 Norm convergence of rectangular summability means
  • 6.2 Almost everywhere restricted summability
  • 6.3 Restricted convergence at Lebesgue points
  • 6.4 Almost everywhere unrestricted summability
  • 6.5 Unrestricted convergence at Lebesgue points
  • Bibliography
  • Index
  • Notations

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