arXiv subjects

Selma Negzaoui

Publications and source records attributed to Selma Negzaoui.

5 recordsLinked to original sources

arXiv2026

Hardy's Theorem for the $(k,\frac{2}{n})-$Fourier Transform

By comparing a function and its $(k, \frac{2}{n})-$Fourier transform to a Gaussian analogue, $e^{-na|x|^\frac{2}{n}}$, we establish a Hardy-type uncertainty principle using Phragmén-Lindlöf lemma. Furthermore, we investigate the heat equation in this context, deriving a dynamical version of Hardy's theorem that illustrates the temporal evolution of the uncertainty principle. We also extend our results to $L^p-L^q$ versions, proving Miyachi-type and Cowling-Price-type theorems for the $(k,\frac{2}{n})$-Fourier transform.

math.CA↗

arXiv2026

A Schwartz-type Space for the $\left(k,\frac{2}{n}\right)-$Generalized Fourier Transform

The Schwartz space $\mathcal{S}(\mathbb{R}^N)$ is not invariant under the $(k,a)$-generalized Fourier transform $\mathcal{F}_{k,a}$ unless $a=2$, and in general no such adapted space is known. For $N=1$ and $\displaystyle a=\frac{2}{n}$, $n\in\mathbb{N}$, we construct a tailored Schwartz-type space $\mathcal{S}_{k,n}(\mathbb{R})$ defined via seminorms built from natural second-order operators associated with the one-dimensional Dunkl Laplacian $Δ_k$. We prove that $\mathcal{S}_{k,n}(\mathbb{R})$ recovers the two basic features of the classical Schwartz space: invariance under the corresponding Fourier-type operator and density in the relevant weighted $L^p-$spaces. To establish these results, we introduce the space $\mathcal{D}_{k,n}(\mathbb{R})$ of compactly supported smooth functions, which embeds continuously into $\mathcal{S}_{k,n}(\mathbb{R})$ and is dense in the weighted spaces $L^p(dμ_{k,n})$, $1\le p<\infty$. These results provide the first Schwartz-type space for $\mathcal{F}_{k,a}$ that simultaneously ensures invariance and $L^p$-density, and admits an $\mathfrak{sl}(2,\mathbb{R})$-based description of the underlying operator structure.

math.CA↗

arXiv2020

A New Product Formula Involving Bessel Functions

In this paper, we consider the normalized Bessel function of index $α> -\frac{1}{2}$, we find an integral representation of the term $x^nj_{α+n}(x)j_α(y)$. This allows us to establish a product formula for the generalized Hankel function $B^{κ,n}_λ$ on $\mathbb{R}$. $B^{κ,n}_λ$ is the kernel of the integral transform $\mathcal{F}_{κ,n}$ arising from the Dunkl theory. Indeed we show that $B^{κ,n}_λ(x)B^{κ,n}_λ(y)$ can be expressed as an integral in terms of $B^{κ,n}_λ(z)$ with explicit kernel invoking Gegenbauer polynomials for all $n\in\mathbb{N}^\ast$. The obtained result generalizes the product formulas proved by M. Rösler for Dunkl kernel when n=1 and by S. Ben Said when $n=2$. \\ As application, we define and study a translation operator and a convolution structure associated to $B^{κ,n}_λ$. They share many important properties with their analogous in the classical Fourier theory.

math.CA↗

arXiv2014

Sonine Transform Associated to the Bessel-Struve Operator

In this paper we consider the Bessel-Struve operator $l_α$ and the Bessel-Struve intertwining operator $χ_α$ and its dual, we define and study the Bessel-Struve Sonine transform $S_{α,β}$ on $\mathcal{E}(\mathbb{R})$. We prove that $S_{α,β}$ is a transmutation operator from $l_α$ into $l_β$ on $\mathcal{E}(\mathbb{R})$ and we deduce similar result for its dual $S_{α,β}^*$ on $\mathcal{E}'(\mathbb{R})$. Furthermore, invoking Weyl integral transform and the Dual Sonine transform $^tS_{α,β}$ on $\mathcal{D}(\mathbb{R})$, we get a relation between the Bessel-Struve transforms $\mathcal{F}^α_{BS} $ and $\mathcal{F}^β_{BS} $.

math.CA↗

arXiv2010

On the Harmonic Analysis Associated to the Bessel-Struve Operator

In this paper, we introduce the Bessel-Struve transform, we establish an inversion theorem of the Weyl integral transform associated with this transform, in the case of half integers, we give a characterization of the range of $\mathcal{D}(\mathbb{R})$ by Bessel-Struve transform and we prove a Schwartz-Paley-Wiener theorem on $\mathcal{E}'(\mathbb{R})$.

math.CA↗