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Safa Snoun

Publications and source records attributed to Safa Snoun.

6 recordsLinked to original sources
arXiv2025

Spectral properties of the Cauchy transform on modified Bergman spaces

In this paper, we determine the singular values $s_n(T_{α,β})$ and $s_n(R_{α,β})$ of the operators $T_{α,β}=\mathcal C\mathbb P_{α,β}$ and $R_{α,β}=\mathbb P_{α,β}\mathcal C\mathbb P_{α,β}$ where $\mathcal C$ is the integral Cauchy transform and $\mathbb P_{α,β}$ is the orthogonal projection from $L^2(\mathbb D,μ_{α,β})$ onto the modified Bergman space $\mathcal A^2(\mathbb D,μ_{α,β})$. These singular values will be expressed in terms of some series involving hypergeometric functions. We show that in both cases the sequence $n^{α+1}s_n(.)$ has a finite limit as $n\to+\infty$.

math.CV↗
arXiv2023

Toeplitz operators via Carleson measures On $β$-modified Bergman Spaces

In this paper, we study Bergman projection $\mathbb{P}_{α,β}$ and Toeplitz operators $T^{α,β}_φ$ on the $β$-modified Bergman space $\mathcal{A}_{α,β}^p$. We give some properties of $\mathbb{P}_{α,β}$ and a necessary and sufficient condition for $T^{α,β}_φ$ to be compact. We end with a characterization of Toeplitz operators via Carleson measures by introducing a new Bergman metric inherited by the Bergman kernel $\mathbb{K}_{α,β}$ that will be equivalent to the classical Bergman-Poincaré metric.

math.CV↗
arXiv2023

Asymptotic results on modified Bergman-Dirichlet spaces and examples of Segal-Bargmann transforms

In this paper, we start by introducing the modified Bergman-Dirichlet space $\mathcal D_m^2(\mathbb D_R,μ^R_{α,β})$ and then we study its asymptotic behavior when the parameter $α$ goes to infinity and to $(-1)$ to obtain respectively the modified Bargmann-Dirichlet and the modified Hardy-Dirichlet spaces with their reproducing kernels. Finally, we give some examples of Segal-Bargmann transforms of those spaces.

math.CV↗
arXiv2023

Fractional Zernike functions

We consider and provide an accurate study for the fractional Zernike functions on the punctured unit disc, generalizing the classical Zernike polynomials and their associated $β$-restricted Zernike functions. Mainly, we give the spectral realization of the latter ones and show that they are orthogonal $L^2$-eigenfunctions for certain perturbed magnetic (hyperbolic) Laplacian. The algebraic and analytic properties for the fractional Zernike functions to be established include the connection to special functions, their zeros, their orthogonality property, as well as the differential equations, recurrence, and operational formulas they satisfy. Integral representations are also obtained. Their regularity as poly-meromorphic functions is discussed and their generating functions including a bilinear one of "Hardy--Hille type" are derived. Moreover, we prove that a truncated subclass defines a complete orthogonal system in the underlying Hilbert space giving rise to a specific Hilbertian orthogonal decomposition in terms of a second class of generalized Bergman spaces.

math.CV↗
arXiv2020

Zeros of new Bergman kernels

In this paper we determine explicitly the kernels $\mathbb K_{α,β}$ associated with new Bergman spaces $\mathcal A_{α,β}^2(\mathbb D)$ considered recently by the first author and M. Zaway. Then we study the distribution of the zeros of these kernels essentially when $α\in\mathbb N$ where the zeros are given by the zeros of a real polynomial $Q_{α,β}$. Some numerical results are given throughout the paper.

math.CV↗