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Allal Ghanmi

Publications and source records attributed to Allal Ghanmi.

At least 19 recordsLinked to original sources
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↗
arXiv2022

Poly-meromorphic Itô-Hermite functions associated with a singular potential vector on the punctured complex plane

We provide a theoretical study of a new family of orthogonal functions on the punctured complex plane solving the eigenvalue problems for some magnetic Laplacian perturbed by a singular vector potential with zero magnetic field modeling the Aharonov-Bohm effect. The functions are defined by their $β$-modified Rodrigues type formula and extend the poly-analytic Itô--Hermite polynomials to the poly-meromorphic setting. Mainly, we derive their different operational representations and give their explicit expressions in terms of special functions. Different generating functions and integral representations are obtained.

math-ph↗
arXiv2022

Weighted quaternionic Cauchy singular integral

We investigate some spectral properties of the weighted quaternionic Cauchy transform when acting on the right quaternionic Hilbert space of Gaussian integrable functions. We study its boundedness, compactness, and memberships to the $k$-Schatten class, and we identify its range. This is done by means of its restriction to the n-th S-polyregular Bargmann space of the second kind, for which we provide an explicit closed expression for its action on the quaternionic Itô--Hermite polynomials constituting an orthogonal basis. We also exhibit an orthogonal basis of eigenfunctions of its n-Bergman projection leading to the explicit determination of its singular values. The obtained results generalize those given for weighted Cauchy transform on the complex plane to the quaternionic setting.

math.CV↗
arXiv2022

Bicomplex polyharmonicity and polyholomorphy

In this paper, we are concerned with the bicomplex analog of the well-known result asserting that real-valued harmonic functions, on simply connected domains, are the real parts of holomorphic functions. We show that this assertion, word for word, fails for bc-harmonic functions and we provide a complete characterization of bc-harmonic functions that are the hyperbolic real parts of a specific kind of bc-holomorphic functions. Moreover, we extend the result to bicomplex polyharmonic functions, which implies the introduction of specific classes of bc-polyholomorphic functions.

math.CV↗
arXiv2021

Two-dimensional $(p,q)$-heat polynomials of Gould--Hopper type

We introduce a new class of holomorphic polynomials extending the classical Gould--Hopper to two complex variables. The considered polynomials include the $1$-D and $2$-D holomorphic and polyanalytic Itô--Hermite polynomials as particular cases. We emphasize studying their operational representation, various generating functions, and recurrence relations. We also establish some special identities including multiplication and addition formulas of Runge type, as well as the Nielson type formulas. Higher-order partial differential equations are analyzed and the connection to Gould-Hopper polynomials and hypergeometric functions are investigated.

math.CA↗
arXiv2020

A lifting theorem for planar mixed automorphic functions and applications

We deal with the concrete spectral analysis of an invariant magnetic Schrödinger operator acting on one dimensional $L^2$-mixed automorphic functions with respect to given equivariant pair $ (ρ,τ) $ and given discrete subgroup of the semi-direct group $U(1)\ltimes\mathbb{C}$. This will be carried out by means of a lifting theorem to the classical automorphic functions associated with specific pseudo-character. We also provided a partial characterization of the equivariant pairs relative to our setting and discuss possible generalization to higher dimensions.

math.SP↗
arXiv2020

$(p,q)$-complex Itô--Hermite polynomials

We introduce two classes of $(p,q)$-Itô--Hermite polynomials, the post-quantum analogs of the $q$-Itô--Hermite polynomials introduced recently by Ismail and Zhang. We study their basic properties such as their operational formulas of Rodrigues type, the corresponding raising and lowering operators as well as their generating functions and the $(p,p$-differential equations they obey.

math.CA↗
arXiv2020

On the range of weighted planar Cauchy transform

We describe the range of of weighted Cauchy transform and its $k$-Bergman projection when action on weighted true poly-Bargmann spaces constituting an orthogonal Hilbertian decomposition of the Hilbert space of Gaussian functions on the complex plane.

math.CV↗
arXiv2020

Dual of 2D fractional Fourier transform associated to Itô--Hermite polynomials

A class of integral transforms, on the planar Gaussian Hilbert space with range in the weighted Bergman space on the bi-disk, is defined as the dual transforms of the 2d fractional Fourier transform associated with the Mehler function for Itô--Hermite polynomials. Some spectral properties of these transforms are investigated. Namely, we study their boundedness and identify their null spaces as well as their ranges. Such identification depends on the zeros set of Itô--Hermite polynomials. Moreover, the explicit expressions of their singular values are given and compactness and membership in p-Schatten class are studied. The relationship to specific fractional Hankel transforms is also established

math.CV↗
arXiv2020

On dual transform of fractional Hankel transform

We deal with a class of one-parameter family of integral transforms of Bargmann type arising as dual transforms of fractional Hankel transform. Their ranges are identified to be special subspaces of the weighted hyperholomorphic left Hilbert spaces, generalizing the slice Bergman space of the second kind. Their reproducing kernel is given by closed expression involving the $\star$-regularization of Gauss hypergeometric function. We also discuss their basic properties such as their boundedness and we determinate their singular values. Moreover, we describe their compactness and membership in $p$-Schatten classes.

math.CV↗
arXiv2020

Bargmann's versus of the quaternionic fractional Hankel transform

We investigate the quaternionic extension of the fractional Fourier transform on the real half-line leading to fractional Hankel transform. This will be handled à la Bargmann by means of hyperholomorphic second Bargmann transform for the slice Bergman space of second kind. Basic properties are derived including inversion formula and Plancherel identity.

math.CV↗
arXiv2020

Bicomplex frames

The main purpose is to introduce the so-called bicomplex (bc)-frames which is a special extension to bicomplex infinite Hilbert spaces of the classical frames. The crucial result is the characterization of bc-frames in terms of their idempotent components, giving rise to generalization of certain results to bc-frames. Although the extension is natural, many basic properties satisfied by classical frames do not remain valid for bc-frames, unless we restrict ourself to complex-valued Hilbert space on bicomplex numbers. By benefiting from insight provided by the classical frame theory, we discuss the construction of bc-frame operator and Weyl--Heisenberg bc-frames and we provide some new ones which are appropriate for bc-frames.

math.FA↗
arXiv2019

Bivariate poly-analytic Hermite polynomials

A new class of bivariate poly-analytic Hermite polynomials is considered. We show that they are realizable as the Fourier-Wigner transform of the univariate complex Hermite functions and form a nontrivial orthogonal basis of the classical Hilbert space on the two-complex space with respect to the Gaussian measure. Their basic properties are discussed, such as their three term recurrence relations, operational realizations and differential equations (Bochner's property) they obey. Different generating functions of exponential type are obtained. Integral and exponential operational representations are also derived. Some applications in the context of integral transforms and the concrete spectral theory of specific magnetic Laplacians are discussed.

math.CV↗
arXiv2019

S--polyregular Bargmann spaces

We introduce two classes of right quaternionic Hilbert spaces in the context of slice polyregular functions, generalizing the so-called slice and full hyperholomorphic Bargmann spaces. Their basic properties are discussed, the explicit formulas of their reproducing kernels are given and associated Segal--Bargmann transforms are also introduced and studied. The spectral description as special subspaces of $L^2$-eigenspaces of a second order differential operator involving the slice derivative is investigated.

math.CV↗
arXiv2019

Bicomplex analogs of Segal-Bargmann and fractional Fourier transforms

We consider and discuss some basic properties of the bicomplex analog of the classical Bargmann space. The explicit expression of the integral operator connecting the complex and bicomplex Bargmann spaces is also given. The corresponding bicomplex Segal--Bargmann transform is introduced and studied as well. Its explicit expression as well as the one of its inverse are then used to introduce a class of two--parameter bicomplex Fourier transforms (bicomplex fractional Fourier transform). This approach is convenient in exploring some useful properties of this bicomplex fractional Fourier transform.

math.CV↗
arXiv2019

The orthogonal complement of the Hilbert space associated to holomorphic Hermite polynomials

We study the orthogonal complement of the Hilbert subspace considered by by van Eijndhoven and Meyers in [J. Math. Anal. Appl. 146 (1990), no. 1, 89--98} and associated to holomorphic Hermite polynomials. A polyanalytic orthonormal basis is given and the explicit expressions of the corresponding reproducing kernel functions and Segal--Bargmann integral transforms are provided.

math.CA↗
arXiv2019

On bicomplex Fourier--Wigner transforms

We consider the $1$- and $2$-d bicomplex analogs of the classical Fourier--Wigner transform. Their basic properties, including Moyal's identity and characterization of their ranges giving rise to new bicomplex--polyanalytic functional spaces are discussed. Particular case of special window is also considered. An orthogonal basis for the space of bicomplex--valued square integrable functions on the bicomplex numbers is constructed by means of the polyanalytic complex Hermite functions.

math.CV↗