Convergence and Summability of Fourier Transforms and Hardy Spaces (eBook)

Ferenc Weisz is Professor at the Department of Numerical Analysis of the Eötvös Loránd University in Budapest, Hungary.

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.