Journal ArticleDOI
The Fourier Transform of a Function of Bounded Variation: Symmetry and Asymmetry
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TLDR
In this article, the cosine Fourier transform of a function of bounded variation and the Hilbert transform of its derivative are revealed, and the obtained results are used for obtaining completely new results on the integrability of trigonometric series.Abstract:
New relations between the Fourier transform of a function of bounded variation and the Hilbert transform of its derivative are revealed. The main result of the paper is an asymptotic formula for the cosine Fourier transform. Such relations have previously been known only for the sine Fourier transform. For this, not only a different space is considered but also a new way of proving such theorems is applied. Interrelations of various function spaces are studied in this context. The obtained results are used for obtaining completely new results on the integrability of trigonometric series.read more
Citations
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Journal ArticleDOI
Fourier analysis with generalized integration
TL;DR: In this article, the Henstock-Kurzweil integral theory was used to generalize the Fourier transform operator to a dense subspace of dimension ρ, where ρ = ρ ∈ ρ 2.
Journal ArticleDOI
Asymptotic decay of Fourier, Laplace and other integral transforms
TL;DR: In this article, the authors studied the asymptotic decay of the cosine and sine Fourier transform, the Hankel transform, and the Laplace transform under appropriate assumptions on the kernels and on the functions involved.
Journal ArticleDOI
Fourier Analysis with Generalized Integration
TL;DR: In this article, the authors generalize the Fourier transform operator F p by using the Henstock-Kurzweil integral theory, and show that the operator equals the H K -Fourier transform on a dense subspace of L p, 1 < p ≤ 2.
Journal ArticleDOI
Fourier transforms on weighted amalgam-type spaces
TL;DR: In this paper, the integrability of trigonometric series with the sequence of coefficients of bounded variation with respect to weighted amalgam-type spaces has been analyzed for the Fourier transform of a function with the derivative from one of those spaces.
Book ChapterDOI
Hardy Spaces and their Subspaces
TL;DR: In this article, it is shown that the Hilbert transform is not necessarily integrable and, in fact, it can be even not locally integrably integrability, even if the Hilbert transformation is locally and integrally coherent with respect to the Hardy space.
References
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Book
Introduction to Fourier Analysis on Euclidean Spaces.
Elias M. Stein,Guido Weiss +1 more
TL;DR: In this paper, the authors present a unified treatment of basic topics that arise in Fourier analysis, and illustrate the role played by the structure of Euclidean spaces, particularly the action of translations, dilatations, and rotations.
Book
Weighted norm inequalities and related topics
TL;DR: Theories de la factorisation and inegalites en norme ponderees of Hardy as mentioned in this paper have been studied in the context of factorization, and a variable reelle des espaces de Hardy has been proposed.
Journal ArticleDOI
Lipschitz Spaces and Bernstein's Theorem on Absolutely Convergent Fourier Transforms*
Book
Fourier Analysis and Approximation of Functions
TL;DR: In this article, the authors present a discussion on representation at a point, including convergence and divergence, convergence in Lp-norm and almost everywhere, and convergence in the space C. The Paley-Wiener theorem, the Chebyshev alternation, and the Wiener Tauberian theorem.