Operational calculus and Fourier transform on Boehmians

V. Karunakaran; R. Roopkumar

Displaying similar documents to “Operational calculus and Fourier transform on Boehmians”

Comparison theorem between Fourier transform and Fourier transform with compact support

Christine Huyghe (2013)

Journal de Théorie des Nombres de Bordeaux

Similarity:

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

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

Similarity:

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.

Uncertainty principles for integral operators

Saifallah Ghobber, Philippe Jaming (2014)

Studia Mathematica

Similarity:

The aim of this paper is to prove new uncertainty principles for integral operators with bounded kernel for which there is a Plancherel Theorem. The first of these results is an extension of Faris’s local uncertainty principle which states that if a nonzero function $f \in L ² (ℝ^{d}, μ)$ is highly localized near a single point then (f) cannot be concentrated in a set of finite measure. The second result extends the Benedicks-Amrein-Berthier uncertainty principle and states that a nonzero function $f \in L ² (ℝ^{d}, μ)$ and...

Absolute convergence of multiple Fourier integrals

Yurii Kolomoitsev, Elijah Liflyand (2013)

Studia Mathematica

Similarity:

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.

On the Hausdorff-Young theorem for commutative hypergroups

Sina Degenfeld-Schonburg (2013)

Colloquium Mathematicae

Similarity:

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.

The Fourier transform in Lebesgue spaces

Erik Talvila (2025)

Czechoslovak Mathematical Journal

Similarity:

For each $f \in L^{p} (ℝ)$ ( $1 \leq p < \infty$ ) it is shown that the Fourier transform is the distributional derivative of a Hölder continuous function. For each $p$ , a norm is defined so that the space of Fourier transforms is isometrically isomorphic to $L^{p} (ℝ)$ . There is an exchange theorem and inversion in norm.

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

Giancarlo Travaglini (1987)

Colloquium Mathematicae

Similarity:

On functions whose translates are independent

Ralph E. Edwards (1951)

Annales de l'institut Fourier

Similarity:

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...

Conditional Fourier-Feynman transform given infinite dimensional conditioning function on abstract Wiener space

Jae Gil Choi, Sang Kil Shim (2023)

Czechoslovak Mathematical Journal

Similarity:

We study a conditional Fourier-Feynman transform (CFFT) of functionals on an abstract Wiener space $(H, B, ν)$ . An infinite dimensional conditioning function is used to define the CFFT. To do this, we first present a short survey of the conditional Wiener integral concerning the topic of this paper. We then establish evaluation formulas for the conditional Wiener integral on the abstract Wiener space $B$ . Using the evaluation formula, we next provide explicit formulas for CFFTs of functionals in...

Sidon sets and $I^{0}$ -sets

David Grow (1987)

Colloquium Mathematicae

Similarity:

On the Hermite expansions of functions from the Hardy class

Rahul Garg, Sundaram Thangavelu (2010)

Studia Mathematica

Similarity:

Considering functions f on ℝⁿ for which both f and f̂ are bounded by the Gaussian $e^{- 1 / 2 a | x | ²}$ , 0 < a < 1, we show that their Fourier-Hermite coefficients have exponential decay. Optimal decay is obtained for O(n)-finite functions, thus extending a one-dimensional result of Vemuri.

Noncommutative extensions of the Fourier transform and its logarithm

Romuald Lenczewski (2002)

Studia Mathematica

Similarity:

We introduce noncommutative extensions of the Fourier transform of probability measures and its logarithm to the algebra (S) of complex-valued functions on the free semigroup S = FS(z,w) on two generators. First, to given probability measures μ, ν with all moments finite, we associate states μ̂, ν̂ on the unital free *-bialgebra (ℬ,ε,Δ) on two self-adjoint generators X,X’ and a projection P. Then we introduce and study cumulants which are additive under the convolution μ̂* ν̂ = μ̂ ⊗...

Bases in spaces of analytic germs

Michael Langenbruch (2012)

Annales Polonici Mathematici

Similarity:

We prove precise decomposition results and logarithmically convex estimates in certain weighted spaces of holomorphic germs near ℝ. These imply that the spaces have a basis and are tamely isomorphic to the dual of a power series space of finite type which can be calculated in many situations. Our results apply to the Gelfand-Shilov spaces $S ¹_{α}$ and $S ₁^{α}$ for α > 0 and to the spaces of Fourier hyperfunctions and of modified Fourier hyperfunctions.

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

Gady Kozma, Alexander Olevskiĭ (2013)

Annales de l’institut Fourier

Similarity:

We study the “Fourier symmetry” of measures and distributions on the circle, in relation with the size of their supports. The main results of this paper are: (i) A one-side extension of Frostman’s theorem, which connects the rate of decay of Fourier transform of a distribution with the Hausdorff dimension of its support; (ii) A construction of compacts of “critical” size, which support distributions (even pseudo-functions) with anti-analytic part belonging to...

On the order of magnitude of Walsh-Fourier transform

Bhikha Lila Ghodadra, Vanda Fülöp (2020)

Mathematica Bohemica

Similarity:

For a Lebesgue integrable complex-valued function $f$ defined on $ℝ^{+} : = [0, \infty)$ let $\hat{f}$ be its Walsh-Fourier transform. The Riemann-Lebesgue lemma says that $\hat{f} (y) \to 0$ as $y \to \infty$ . But in general, there is no definite rate at which the Walsh-Fourier transform tends to zero. In fact, the Walsh-Fourier transform of an integrable function can tend to zero as slowly as we wish. Therefore, it is interesting to know for functions of which subclasses of $L^{1} (ℝ^{+})$ there is a definite rate at which the Walsh-Fourier transform tends...

On the diametral dimension of weighted spaces of analytic germs

Michael Langenbruch (2016)

Studia Mathematica

Similarity:

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.

Restriction theorems for the Fourier transform to homogeneous polynomial surfaces in ℝ³

E. Ferreyra, T. Godoy, M. Urciuolo (2004)

Studia Mathematica

Similarity:

Let φ:ℝ² → ℝ be a homogeneous polynomial function of degree m ≥ 2, let Σ = (x,φ(x)): |x| ≤ 1 and let σ be the Borel measure on Σ defined by $σ (A) = \int_{B} χ_{A} (x, φ (x)) d x$ where B is the unit open ball in ℝ² and dx denotes the Lebesgue measure on ℝ². We show that the composition of the Fourier transform in ℝ³ followed by restriction to Σ defines a bounded operator from $L^{p} (ℝ ³)$ to $L^{q} (Σ, d σ)$ for certain p,q. For m ≥ 6 the results are sharp except for some border points.

A necessary condition for HK-integrability of the Fourier sine transform function

Juan H. Arredondo, Manuel Bernal, Maria G. Morales (2025)

Czechoslovak Mathematical Journal

Similarity:

The paper is concerned with integrability of the Fourier sine transform function when $f \in {BV}_{0} (ℝ)$ , where ${BV}_{0} (ℝ)$ is the space of bounded variation functions vanishing at infinity. It is shown that for the Fourier sine transform function of $f$ to be integrable in the Henstock-Kurzweil sense, it is necessary that $f / x \in L^{1} (ℝ)$ . We prove that this condition is optimal through the theoretical scope of the Henstock-Kurzweil integration theory.

A variation norm Carleson theorem

Richard Oberlin, Andreas Seeger, Terence Tao, Christoph Thiele, James Wright (2012)

Journal of the European Mathematical Society

Similarity:

We strengthen the Carleson-Hunt theorem by proving $L^{p}$ estimates for the $r$ -variation of the partial sum operators for Fourier series and integrals, for $r > 𝚖𝚊𝚡 {p^{'}, 2}$ . Four appendices are concerned with transference, a variation norm Menshov-Paley-Zygmund theorem, and applications to nonlinear Fourier transforms and ergodic theory.

The zeros of functions of finite order in $C^{n}$

Lawrence Gruman (1983)

Annales Polonici Mathematici

Similarity:

Best constants for some operators associated with the Fourier and Hilbert transforms

B. Hollenbeck, N. J. Kalton, I. E. Verbitsky (2003)

Studia Mathematica

Similarity:

We determine the norm in $L^{p} (ℝ ₊)$ , 1 < p < ∞, of the operator $I - ℱ_{s} ℱ_{c}$ , where $ℱ_{c}$ and $ℱ_{s}$ are respectively the cosine and sine Fourier transforms on the positive real axis, and I is the identity operator. This solves a problem posed in 1984 by M. S. Birman [Bir] which originated in scattering theory for unbounded obstacles in the plane. We also obtain the $L^{p}$ -norms of the operators aI + bH, where H is the Hilbert transform (conjugate function operator) on the circle or real line, for arbitrary real...