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 generalized convolution with a weight function for the Fourier cosine and sine transforms is introduced. Its properties and applications to solving a system of integral equations are considered.

The paper is concerned with integrability of the Fourier sine transform function when $f\in {\mathrm{BV}}_0\left(ℝ\right)$, where ${\mathrm{BV}}_0\left(ℝ\right)$ 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 $f/x\in L^1\left(ℝ\right)$. We prove that this condition is optimal through the theoretical scope of the Henstock-Kurzweil integration theory.

For a fusion Banach frame $(\{G_n,v_n\},S)$ for a Banach space $E$, if $(\{v_n^{*}\left(E^{*}\right),v_n^{*}\},T)$ is a fusion Banach frame for $E^{*}$, then $(\{G_n,v_n\},S;\{v_n^{*}\left(E^{*}\right),v_n^{*}\},T)$ 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.

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

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 smooth interpolation 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...

Let $\mu$ 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 $\mathbf{R}$. Then for every $ϵ\>0$, $\{x\in \mathbf{R}|\mathrm{Re}\left(\mu \right(x\left)\right)\>ϵ\}$ 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 general method is given for recovering a function $f:{\mathbb{R}}^N\to C$, $N\ge 1$, knowing only an approximation of its Fourier transform.

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.

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.