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A characterization of Fourier transforms

Philippe Jaming (2010)

Colloquium Mathematicae

The aim of this paper is to show that, in various situations, the only continuous linear (or not) map that transforms a convolution product into a pointwise product is a Fourier transform. We focus on the cyclic groups ℤ/nℤ, the integers ℤ, the torus 𝕋 and the real line. We also ask a related question for the twisted convolution.

A note on fusion Banach frames

S. K. Kaushik, Varinder Kumar (2010)

Archivum Mathematicum

For a fusion Banach frame ( { G n , v n } , S ) for a Banach space E , if ( { v n * ( E * ) , v n * } , T ) is a fusion Banach frame for E * , then ( { G n , v n } , S ; { v n * ( E * ) , v n * } , T ) is called a fusion bi-Banach frame for E . It is proved that if E has an atomic decomposition, then E also has a fusion bi-Banach frame. Also, a sufficient condition for the existence of a fusion bi-Banach frame is given. Finally, a characterization of fusion bi-Banach frames is given.

A note on the convolution theorem for the Fourier transform

Charles S. Kahane (2011)

Czechoslovak Mathematical Journal

In this paper we characterize those bounded linear transformations T f carrying L 1 ( 1 ) into the space of bounded continuous functions on 1 , for which the convolution identity T ( f * g ) = T f · T g holds. It is shown that such a transformation is just the Fourier transform combined with an appropriate change of variable.

A particular smooth interpolation that generates splines

Segeth, Karel (2017)

Programs and Algorithms of Numerical Mathematics

There are two grounds the spline theory stems from – the algebraic one (where splines are understood as piecewise smooth functions satisfying some continuity conditions) and the variational one (where splines are obtained via minimization of some quadratic functionals with constraints). We use the general variational approach called 𝑠𝑚𝑜𝑜𝑡ℎ𝑖𝑛𝑡𝑒𝑟𝑝𝑜𝑙𝑎𝑡𝑖𝑜𝑛 introduced by Talmi and Gilat and show that it covers not only the cubic spline and its 2D and 3D analogues but also the well known tension spline (called also spline...

A property of Fourier Stieltjes transforms on the discrete group of real numbers

Yngve Domar (1970)

Annales de l'institut Fourier

Let μ be a Fourier-Stieltjes transform, defined on the discrete real line and such that the corresponding measure on the dual group vanishes on the set of characters, continuous on R . Then for every ϵ > 0 , { x R | Re ( μ ( x ) ) > ϵ } has a vanishing interior Lebesgue measure. If ϵ = 0 the statement is not generally true. The result is applied to prove a theorem of Rosenthal.

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

A general method is given for recovering a function f : R N C , N 1 , knowing only an approximation of its Fourier transform.

A Tauberian theorem for distributions

Jiří Čížek, Jiří Jelínek (1996)

Commentationes Mathematicae Universitatis Carolinae

The well-known general Tauberian theorem of N. Wiener is formulated and proved for distributions in the place of functions and its Ganelius' formulation is corrected. Some changes of assumptions of this theorem are discussed, too.

Absolute convergence of multiple Fourier integrals

Yurii Kolomoitsev, Elijah Liflyand (2013)

Studia Mathematica

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.

Almost everywhere convergence of convolution powers on compact abelian groups

Jean-Pierre Conze, Michael Lin (2013)

Annales de l'I.H.P. Probabilités et statistiques

It is well-known that a probability measure μ on the circle 𝕋 satisfies μ n * f - f d m p 0 for every f L p , every (some) p [ 1 , ) , if and only if | μ ^ ( n ) | l t ; 1 for every non-zero n ( μ is strictly aperiodic). In this paper we study the a.e. convergence of μ n * f for every f L p whenever p g t ; 1 . We prove a necessary and sufficient condition, in terms of the Fourier–Stieltjes coefficients of μ , for the strong sweeping out property (existence of a Borel set B with lim sup μ n * 1 B = 1 a.e. and lim inf μ n * 1 B = 0 a.e.). The results are extended to general compact Abelian groups G with Haar...

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