Related papers: Fourier inversion theorems for integral transforms…

This is the first part of a series of papers where the behaviour of the invariants under twist by Dirichlet characters is studied for $L$-functions of degree 2. Here we show, under suitable conditions, that degree and internal shift remain…

Spectral decomposition of dynamical equations using curl-eigenfunctions has been extensively used in fluid and plasma dynamics problems using their orthogonality and completeness properties for both linear and non-linear cases. Coefficients…

This comprehensive review paper delves into the intricacies of advanced Fourier type integral transforms and their mathematical properties, with a particular focus on fractional Fourier transform (FrFT), linear canonical transform (LCT),…

The classical Fourier transform is, in essence, a way to take data and extract components (in the form of complex exponentials) which are invariant under cyclic shifts. We consider a case in which the components must instead be invariant…

We discuss various forms of the Plancherel Formula and the Plancherel Theorem on reductive groups over local fields.

We consider the image of the operator inducing the determinantal point process with the confluent hypergeometric kernel. The space is described as the image of $L_2[0, 1]$ under a unitary transform, which generalizes the Fourier transform.…

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…

We consider an integral transform given by $T_{\nu} f(s) := \pi \int_0^\infty rs J_{\nu}(r s)^2 f(r) \, dr$, where $J_{\nu}$ denotes the Bessel function of the first kind of order $\nu$. As shown by Walther (2002,…

In this paper, we introduce a family of integral transforms, denoted by \(\mathcal{O}_{\alpha}\), and constructed via kernel fusion of the fractional Fourier transform (FRFT) with angle \(\alpha \notin \pi \mathbb{Z}\). We demonstrate that…

We treat the quaternionic Fourier transform (QFT) applied to quaternion fields and investigate QFT properties useful for applications. Different forms of the QFT lead us to different Plancherel theorems. We relate the QFT computation for…

In this paper, a fractional version of the Clifford-Fourier transform is introduced, depending on two numerical parameters. A series expansion for the kernel of the resulting integral transform is derived. In the case of even dimension,…

Shape-invariant signals under Fourier transform are investigated leading to a class of eigenfunctions for the Fourier operator. The classical uncertainty Gabor-Heisenberg principle is revisited and the concept of isoresolution in joint…

We derive inversion formulas involving orthogonal polynomials which can be used to find coefficients of differential equations satisfied by certain generalizations of the classical orthogonal polynomials. As an example we consider special…

A different application of the familiar integral representation for the modifed Bessel function drives to a new Kontorovich-Lebedev-like integral transformation of a general complex index. Mapping and operational properties, a convolution…

We obtain necessary and sufficient conditions on weights for the generalized Fourier-type transforms to be bounded between weighted $L^p-L^q$ spaces. As an important example, we investigate transforms with kernel of power type, as for…

There have been many proposed forms of fractional calculus, which can be grouped into a few broad classes of operators. By replacing the kernel of the power function with another kernel function, the traditional Riemann-Liouville formula…

Analytic expressions for the Fourier transforms of the Chebyshev and Legendre polynomials are derived, and the latter is used to find a new representation for the half-order Bessel functions. The numerical implementation of the so-called…

This paper presents a family of Fourier eigenfunctions indexed by the space dimension d. These eigenfunctions are radial and built upon some generalized exponential integral function. For d=1,2,3, they are integrable or square integrable…

An essential generalization of the Lebedev index transform with the square of the Macdonald function is investigated. Namely, we consider a family of integral operators with the positive kernel $|K_{(i\tau+\alpha)/2}(x)|^2, \alpha \ge 0,\ x…

We explore the spectral properties of the time-dependent Maxwell's equations for a plane interface between a metamaterial represented by the Drude model and the vacuum, which fill respectively complementary half-spaces. We construct…