Singular distributions, dimension of support, and symmetry of Fourier transform

Gady Kozma; Alexander Olevskiĭ

Displaying similar documents to “Singular distributions, dimension of support, and symmetry of Fourier transform”

Operational calculus and Fourier transform on Boehmians

V. Karunakaran, R. Roopkumar (2005)

Colloquium Mathematicae

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We define various operations on the space of ultra Boehmians like multiplication with certain analytic functions which are Fourier transforms of compactly supported distributions, polynomials, and characters $(e^{i s t}, s, t \in ℝ)$ , translation, differentiation. We also prove that the Fourier transform on the space of ultra Boehmians has all the operational properties as in the classical theory.

Comparison theorem between Fourier transform and Fourier transform with compact support

Christine Huyghe (2013)

Journal de Théorie des Nombres de Bordeaux

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We prove a comparison theorem between Fourier transform without support and and Fourier transform with compact support in the context of arithmetic $𝒟$ -modules.

Absolute convergence of multiple Fourier integrals

Yurii Kolomoitsev, Elijah Liflyand (2013)

Studia Mathematica

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Various new sufficient conditions for representation of a function of several variables as an absolutely convergent Fourier integral are obtained. The results are given in terms of $L_{p}$ integrability of the function and its partial derivatives, each with a different p. These p are subject to certain relations known earlier only for some particular cases. Sharpness and applications of the results obtained are also discussed.

Functions which are Fourier transforms of distributions

Raimond A. Struble (1981)

Colloquium Mathematicae

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A stable method for the inversion of the Fourier transform in $R^{N}$

Leonede De Michele, Delfina Roux (1993)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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A general method is given for recovering a function $f : R^{N} \to C$ , $N \geq 1$ , knowing only an approximation of its Fourier transform.

Fourier transforms in generalized Fock spaces.

Schmeelk, John (1990)

International Journal of Mathematics and Mathematical Sciences

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On functions whose translates are independent

Ralph E. Edwards (1951)

Annales de l'institut Fourier

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Ce travail est l’étude de divers cas particuliers d’un problème nouveau, semble-t-il, concernant les translatées de fonctions ou de distributions sur un groupe. Soit $E$ un espace vectoriel topologique de fonctions ou de distributions sur un groupe abélien $G$ localement compact ; $E$ est supposé invariant par les translations $a \to f_{a} (x) = f (x + a) (f \in E, a \in G)$ . Si $f \in E$ et si $A$ est un sous-ensemble non vide de $G$ , $I (f, A) = I (f, A, E)$ désigne le sous-espace vectoriel fermé de $E$ engendré par les translatées $f_{a}$ de $f$ avec $a \in A$ . On dira qu’une $f \in E$ a ses...

Rearrangement of coefficients of Fourier series on $S U_{2}$

Giancarlo Travaglini (1987)

Colloquium Mathematicae

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On the Hausdorff-Young theorem for commutative hypergroups

Sina Degenfeld-Schonburg (2013)

Colloquium Mathematicae

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We study the Hausdorff-Young transform for a commutative hypergroup K and its dual space K̂ by extending the domain of the Fourier transform so as to encompass all functions in $L^{p} (K, m)$ and $L^{p} (K ̂, π)$ respectively, where 1 ≤ p ≤ 2. Our main theorem is that those extended transforms are inverse to each other. In contrast to the group case, this is not obvious, since the dual space K̂ is in general not a hypergroup itself.

On the Fourier transform, Boehmians, and distributions

Dragu Atanasiu, Piotr Mikusiński (2007)

Colloquium Mathematicae

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We introduce some spaces of generalized functions that are defined as generalized quotients and Boehmians. The spaces provide simple and natural frameworks for extensions of the Fourier transform.

On Fourier transforms of distributions with one-sided bounded carriers

Z. Zieleźny (1964)

Studia Mathematica

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The support of a function with thin spectrum

Kathryn Hare (1994)

Colloquium Mathematicae

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We prove that if $E \subseteq Ĝ$ does not contain parallelepipeds of arbitrarily large dimension then for any open, non-empty $S \subseteq G$ there exists a constant c > 0 such that $∥ f 1_{S} ∥_{2} \geq c ∥ f ∥_{2}$ for all $f \in L^{2} (G)$ whose Fourier transform is supported on E. In particular, such functions cannot vanish on any open, non-empty subset of G. Examples of sets which do not contain parallelepipeds of arbitrarily large dimension include all Λ(p) sets.

On the diametral dimension of weighted spaces of analytic germs

Michael Langenbruch (2016)

Studia Mathematica

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We prove precise estimates for the diametral dimension of certain weighted spaces of germs of holomorphic functions defined on strips near ℝ. This implies a full isomorphic classification for these spaces including the Gelfand-Shilov spaces $S ¹_{α}$ and $S ₁^{α}$ for α > 0. Moreover we show that the classical spaces of Fourier hyperfunctions and of modified Fourier hyperfunctions are not isomorphic.

Inverse Fourier transform

Leonede De Michele, Marina Di Natale, Delfina Roux (1990)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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In this paper a very general method is given in order to reconstruct a periodic function $f$ knowing only an approximation of its Fourier coefficients.