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.

For 1 ≤ p,q ≤ ∞, we prove that the convolution operator generated by the Cantor-Lebesgue measure on the circle is a contraction whenever it is bounded from $L^p\left(\right)$ to $L^q\left(\right)$. We also give a condition on p which is necessary if this operator maps $L^p\left(\right)$ into L²().

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 q-convolution is a measure-preserving transformation which originates from non-commutative probability, but can also be treated as a one-parameter deformation of the classical convolution. We show that its commutative aspect is further certified by the fact that the q-convolution satisfies all of the conditions of the generalized convolution (in the sense of Urbanik). The last condition of Urbanik's definition, the law of large numbers, is the crucial part to be proved and the non-commutative...

Soient $G_1$ et $G_2$ deux groupes abéliens localement compacts de dual ${\Gamma }_1$ et ${\Gamma }_2$. Soit $h:{\Gamma }_1\to {\Gamma }_2$ un homomorphisme continu d’image dense de ${\Gamma }_1$ dans ${\Gamma }_2$. Soit $1\le p\le \infty$ ; on prouve un théorème d’approximation des multiplicateurs de $\mathbf{F}{L}^p(G_2)$ et on utilise ce résultat pour démontrer le suivant : soit $m:{\Gamma }_2\to \mathbf{C}$ une fonction continue ; $m$ est un multiplicateur de $\mathbf{F}{L}^p(G_2)$ si, et seulement si, $m\circ h$ est un multiplicateur de $\mathbf{F}{L}^p(G_1)$.