Related papers: The Fourier and Hilbert transforms under the Bargm…
There is a canonical unitary transformation from $L^2(\R)$ onto the Fock space $F^2$, called the Bargmann transform. The purpose of this article is to translate some important results and operators from the context of $L^2(\R)$ to that of…
In this paper we introduce and study a two-parameter family of integral operators on the Fock space $F^2(C)$. We determine exactly when these operators are bounded and when they are unitary. We show that, under the Bargmann transform, these…
We investigate the quaternionic extension of the fractional Fourier transform on the real half-line leading to fractional Hankel transform. This will be handled \`a la Bargmann by means of hyperholomorphic second Bargmann transform for the…
We introduce the Segal-Bargmann transform associated to the Mittag Leffler Fock space and study how it will be connected to the Fourier transform. We will discuss also the counterpart of the creation and annihilation operator in this…
We construct operator analogues of Hermite functions which form an orthonormal basis for the Hilbert space $ \mathcal{S}_2$ of Hilbert-Schmidt operators on $ L^2(\R^n).$ We use this orthonormal basis to define Fourier transform on $…
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…
We study the Beurling and Fourier transforms on subspaces of $L^2({\mathbb C})$ defined by an invariance property with respect to the root-of-unity group. This leads to generalizations of these transformations acting unitarily on weighted…
We use the classical Fourier analysis to introduce analytic families of weighted differential operators on the unit sphere. These operators are polynomial functions of the usual Beltrami-Laplace operator. New inversion formulas are obtained…
The introduction of operator states and of observables in various fields of quantum physics has raised questions about the mathematical structures of the corresponding spaces. In the framework of third quantization it had been conjectured…
We study concentration operators acting on the Fourier symmetric Sobolev space $H$ consisting of functions $f$ such that $\int_{\mathbb{R}} |f(x)|^2(1+x^2) dx + \int_{\mathbb{R}} |\hat{f}(\xi)|^2(1+\xi^2) d\xi < \infty $. We find that the…
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…
Algebraic Bargmann and Darboux transformations for equations of a more general form than the Schr\"odinger ones with an additional functional dependence h(r) in the right-hand side of equations are constructed. The suggested generalized…
Let F(R^n) be the algebra of Fourier transforms of functions from L_1(R^n), K(R^n) be the algebra of Fourier transforms of bounded complex Borel measures in R^n and W be Wiener algebra of continuous 2pi-periodic functions with absolutely…
We consider Fourier transform of vector-valued functions on a locally compact group $G$, which take value in a Banach space $X$, and are square-integrable in Bochner sense. If $G$ is a finite group then Fourier transform is a bounded…
We consider the imaginary time flow of a quadratic hyperbolic Hamiltonian on the symplectic plane, apply it to the Schr\"odinger polarization and study the corresponding evolution of polarized sections. The flow is periodic in imaginary…
The fractional Fourier transform (FrFT), which is a generalization of the Fourier transform, has become the focus of many research papers in recent years because of its applications in electrical engineering and optics. In this paper, we…
We study fractional differential equations of Riemann-Liouville and Caputo type in Hilbert spaces. Using exponentially weighted spaces of functions defined on $\mathbb{R}$, we define fractional operators by means of a functional calculus…
Fourier transforms are ubiquitous mathematical tools in basic and applied sciences. We here report classical and quantum optical realizations of the discrete fractional Fourier transform, a generalization of the Fourier transform. In the…
We introduce the linear operators of fractional integration and fractional differentiation in the framework of the Riemann-Liouville fractional calculus. Particular attention is devoted to the technique of Laplace transforms for treating…
The transformation property of the Caputo fractional derivative operator of a scalar function under rotation in two dimensional space is derived. The study of the transformation property is essential for the formulation of fractional…