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Allal Ghanmi

Publications and source records attributed to Allal Ghanmi.

At least 37 records · Page 2Linked to original sources
arXiv2019

On a novel class of polyanalytic Hermite polynomials

We carry out some algebraic and analytic properties of a new class of orthogonal polyanalytic polynomials, including their operational formulas, recurrence relations, generating functions, integral representations and different orthogonality identities. We establish their connection and rule in describing the $L^2$--spectral theory of some special second order differential operators of Laplacian type acting on the $L^2$--gaussian Hilbert space on the whole complex plane. We will also show their importance in the theory of the so-called rank--one automorphic functions on the complex plane. In fact, a variant subclass leads to an orthogonal basis of the corresponding $L^2$--gaussian Hilbert space on the strip.

math.CV↗
arXiv2019

Polyregularity of the dot product of slice regular functions

In this paper, we are concerned with the S-polyregularity the regular dot product of slice regular functions. We prove that the product of a slice regular function and right quaternionic polynomial function is a S-polyregular function and we determinate its exact order. The general case of the product of any two slice regular functions is also discussed. In fact, we provide sufficient and necessary conditions to the product of slice regular functions be a S-polyregular function. The obtained results are then extended to the product of S-polyregular functions and remain valid for a special dot product. As consequences we obtain linearization theorems for such S-polyregular products with respect to the slice regular functions.

math.CV↗
arXiv2018

On some analytic properties of slice poly-regular Hermite polynomials

We consider a quaternionic analogue of the univariate complex Hermite polynomials and study some of their analytic properties in some detail. We obtain their integral representation as well as the operational formulas of exponential and Burchnall types they obey. We show that they form an orthogonal basis of both the slice and the full Hilbert spaces on the quaternions with respect to the Gaussian measure. We also provide different types of generating functions. Remarkably identities, including quadratic recurrence formulas of Nielsen type are derived.

math.CV↗
arXiv2018

On composition of Segal-Bargmann transforms

We introduce and discuss some basic properties of some integral transforms in the framework of specific functional Hilbert spaces, the holomorphic Bargmann-Fock spaces on $\mathbb{C}$ and $\mathbb{C}^2$ and the slice hyperholomorphic Bargmann-Fock space on $\mathbb{H}$. The first one is a natural integral transform mapping isometrically the standard Hilbert space on the real line into the two-dimensional Bargmann-Fock space. It is obtained as composition of the one and two dimensional Segal-Bargmann transforms and reduces further to an extremely integral operator that looks like a composition operator of the one-dimensional Segal-Bargmann transform with a specific symbol. We study its basic properties, including the identification of its image and the determination of a like-left inverse defined on the whole two-dimensional Bargmann-Fock space. We also examine their combination with the Fourier transform which lead to special integral transforms connecting the two-dimensional Bargmann-Fock space and its analogue on the complex plane. We also investigate the relationship between special subspaces of the two-dimensional Bargmann-Fock space and the slice-hyperholomorphic one on the quaternions by introducing appropriate integral transforms. We identify their image and their action on the reproducing kernel.

math.CV↗
arXiv2018

Non-trivial 1d and 2d integral transforms of Segal-Bargmann type

Generating functions for the univariate complex Hermite polynomials (UCHP) are employed to introduce some non-trivial one and two-dimensional integral transforms of Segal-Bargmann type in the framework of specific functional Hilbert spaces. The approach used is issued from the coherent states framework. Basic properties of these transforms are studied. Connection to some special known transforms like Fourier and Wigner transforms is established. The first transform is a two-dimensional Segal-Bargmann transform whose kernel is related the exponential generating function of the UCHP. The second one connects the so-called generalized Bargmann-Fock spaces (or also true-poly-Fock spaces in the terminology of Vasilevski) that are realized as the $L^2$-eigenspaces of a specific magnetic Schrödinger operator. Its kernel function is related to a Mehler formula of the UCHP.

math.CV↗
arXiv2017

Mehler's formulas for the univariate complex Hermite polynomials and applications

We give two widest Mehler's formulas for the univariate complex Hermite polynomials $H_{m,n}^ν$, by performing double summations involving the products $u^m H_{m,n}^ν(z,\overline{z}) \overline{H_{m,n}^ν(w,\overline{w})}$ and $u^m v^n H_{m,n}^ν(z,\overline{z}) \overline{H_{m,n}^{ν'} (w,\overline{w})}$. They can be seen as the complex analogues of the classical Mehler's formula for the real Hermite polynomials. The proof of the first one is based on a generating function giving rise to the reproducing kernel of the generalized Bargmann space of level $m$. The second Mehler's formula generalizes the one appearing as a particular case of the so-called Kibble-Slepian formula. The proofs, we present here are direct and more simpler. Moreover, direct applications are given and remarkable identities are derived.

math.CA↗
arXiv2017

On concrete spectral properties of a twisted-Laplacian associated to a central extension of the real Heisenberg group

We consider the magnetic Laplacian $Δ_{ν,μ}$ on $\mathbb{R}^{2n}=\mathbb{C}^n$ given by $$ Δ_{ν,μ}= 4\sum\limits_{j=1}\limits^{n}\frac{\partial^2 }{\partial z_j \partial \overline{z_j}} +2iν(E+ \overline{E} +n) +2μ(E- \overline{E} ) -(ν^2+μ^2)|z|^2. $$ We show that $Δ_{ν,μ}$ is connected to the sub-Laplacian of a group of Heisenberg type given by $\mathbb{C}\times_ω\mathbb{C}^n$ realized as a central extension of the real Heisenberg group $H_{2n+1}$. We also discuss invariance properties of $Δ_{ν,μ}$ and give some of their explicit spectral properties.

math.CV↗
arXiv2015

Complex Hermite functions as Fourier-Wigner transform

We prove that the complex Hermite polynomials H_{m,n} on the complex plane $\mathbb{C}$ can be realized as the Fourier-Wigner transform $\mathcal{V}$ of the well-known real Hermite functions $h_n$ on real line $\mathbb{R}$. This reduces considerably the Wong's proof giving the explicit expression of $\mathcal{V}(h_m,h_n)$ in terms of the Laguerre polynomials. Moreover, we derive a new generating function for the H_{m,n} as well as some new integral identities.

math.CA↗
arXiv2015

Weighted Bergman-Dirichlet and Bargmann-Dirichlet spaces of order $m$ in high dimensions

We introduce and study a generalization of the classical weighted Bergman and Dirichlet spaces on the unit ball in high dimension, the Bergman-Dirichlet spaces. Their counterparts on the whole $n$-complex space, the Bargmann-Dirichlet spaces, are also introduced and studied. Mainly, we give a complete description of the considered spaces, including orthonormal basis and the explicit formulas for their reproducing kernel functions. Moreover, we investigate their asymptotic behavior when the curvature goes to $0$.

math.CV↗
arXiv2015

On holomorphic theta functions associated to rank $r$ isotropic discrete subgroups of a $g$-dimensional complex space

We are interested in the $L^2$-holomorphic automorphic functions on a $g$-dimensional complex space $V^g_{\mathbb{C}}$ endowed with a positive definite hermitian form and associated to isotropic discrete subgroups $Γ$ of rank $2\leq r \leq g$. We show that they form an infinite reproducing kernel Hilbert space which looks like a tensor product of a theta Fock-Bargmann space on $V^{r}_{\mathbb{C}}=Span_{\mathbb{C}}(Γ)$ and the classical Fock-Bargmann space on $V^{g-r}_{\mathbb{C}}$. Moreover, we provide an explicit orthonormal basis using Fourier series and we give the expression of its reproducing kernel function in terms of Riemann theta function of several variables with special characteristics.

math.CV↗
arXiv2015

Generalized Zernike polynomials: Operational formulae and generating functions

We establish new operational formulae of Burchnall type for the complex disk polynomials (generalized Zernike polynomials). We then use them to derive some interesting identities involving these polynomials. In particular, we establish recurrence relations with respect to the argument and the integer indices, as well as Nielsen identities and Runge addition formula. In addition, various new generating functions for these disk polynomials are proved.

math.CA↗
arXiv2013

Operational formulae for the complex Hermite polynomials

We give operational formulae of Burchnall type involving complex Hermite polynomials. Short proofs of some known formulae are given and new results involving these polynomials, including Nielsen's identities and Runge addition formula, are derived.

math.CA↗
arXiv2012

Construction of concrete orthonormal basis for (L^2,Γ,χ)-theta functions associated to discrete subgroups of rank one in (C,+)

Let χbe a character on a discrete subgroup Γof rank one of the additive group (C,+). We construct a complete orthonormal basis of the Hilbert space of (L^2,Γ,χ)-theta functions. Furthermore, we show that it possesses a Hilbertian orthogonal decomposition involving the L^2-eigenspaces of the Landau operator Δ_ν; ν>0, associated to the eigenvalues νm. For m=0, the associated L^2-eigenspace is the Hilbert subspace of entire (L^2,Γ,χ)-theta functions. Corresponding orthonormal basis are constructed and the corresponding reproducing kernel can be expressed in terms of the generalized theta function of characteristic [α,0].

math.CV↗
arXiv2012

On a class of two-index real Hermite polynomials

We introduce a class of doubly indexed real Hermite polynomials and we deal with their related properties like the associated recurrence formulae, Runge's addition formula, generating function and Nielsen's identity.

math.CA↗
arXiv2010

On $L^2$-eigenfunctions of Twisted Laplacian on curved surfaces and suggested orthogonal polynomials

We show in a unified manner that the factorization method describes completely the $L^2$-eigenspaces associated to the discrete part of the spectrum of the twisted Laplacian on constant curvature Riemann surfaces. Subclasses of two variable orthogonal polynomials are then derived and arise by successive derivations of elementary complex valued functions depending on the geometry of the surface.

math.SP↗