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 $\mathbf{F}{L}^p(G_1)$.

Extending previous work by Meise and Vogt, we characterize those convolution operators, defined on the space ${}_{\left(\omega \right)}\left(ℝ\right)$ of (ω)-quasianalytic functions of Beurling type of one variable, which admit a continuous linear right inverse. Also, we characterize those (ω)-ultradifferential operators which admit a continuous linear right inverse on ${}_{\left(\omega \right)}[a,b]$ for each compact interval [a,b] and we show that this property is in fact weaker than the existence of a continuous linear right inverse on ${}_{\left(\omega \right)}\left(ℝ\right)$.

Let X be a rearrangement-invariant space of Lebesgue-measurable functions on ${ℝ}^n$, such as the classical Lebesgue, Lorentz or Orlicz spaces. Given a nonnegative, measurable (weight) function on ${ℝ}^n$, define $X\left(w\right)=F:{ℝ}^n\to ℂ:\infty >\parallel F{\parallel }_{X\left(w\right)}:=\parallel Fw{\parallel }_X$. We investigate conditions on such a weight w that guarantee X(w) is an algebra under the convolution product F∗G defined at $x\in {ℝ}^n$ by $\left(F\ast G\right)\left(x\right)={ʃ}_{{ℝ}^n}F(x-y)G\left(y\right)dy$; more precisely, when $\parallel F\ast G{\parallel }_{X\left(w\right)}\le \parallel F{\parallel }_{X\left(w\right)}\parallel G{\parallel }_{X\left(w\right)}$ for all F,G ∈ X(w).

Two important examples of q-deformed commutativity relations are: aa* - qa*a = 1, studied in particular by M. Bożejko and R. Speicher, and ab = qba, studied by T. H. Koornwinder and S. Majid. The second case includes the q-normality of operators, defined by S. Ôta (aa* = qa*a). These two frameworks give rise to different convolutions. In particular, in the second scheme, G. Carnovale and T. H. Koornwinder studied their q-convolution. In the present paper we consider another convolution of measures...