Convergence and Summability of Fourier Transforms and Hardy Spaces

• Κύριος συγγραφέας: Weisz, Ferenc (Συγγραφέας)
• Συγγραφή απο Οργανισμό/Αρχή: SpringerLink (Online service)
• Μορφή: Ηλεκτρονική πηγή Ηλ. βιβλίο
• Γλώσσα: English
• Έκδοση: Cham : Springer International Publishing : Imprint: Birkhäuser, 2017.
• Σειρά: Applied and Numerical Harmonic Analysis,
• Θέματα: Mathematics.; Harmonic analysis.; Fourier analysis.; Sequences (Mathematics).; Sequences, Series, Summability.; Fourier Analysis.; Abstract Harmonic Analysis.

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...

Πίνακας περιεχομένων:

• 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
• 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 `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
• 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.