Related papers: On the radially deformed Fourier transform
We apply the Fourier transform technique and a modified version of E. Stein's interpolation theorem communicated by L. Grafakos, to obtain sharp $L^p$-$L^q$ estimates for the Radon transform and more general convolution-type fractional…
We introduce a deformation of the Fourier transform on $\mathbb{R}^N$ arising from a representation-theoretic construction associated with $\widetilde{SL}(2,\mathbb{R}) \times O(N)$ that still admits an underlying degree-one operator…
The reproducing kernel function of a weighted Bergman space over domains in ${\mathbb C}^d$ is known explicitly in only a small number of instances. Here, we introduce a process of orthogonal norm expansion along a subvariety of codimension…
This work is motivated by the frequent occurrence of boundary value problems with various boundary conditions in the modeling of some problems in engineering and physical science. Here we propose a new technique to force the positive…
In this paper we examine the existence of bicomplexified inverse Fourier transform as an extension of its complexified inverse version within the region of convergence of bicomplex Fourier transform. In this paper we use the idempotent…
By using the three-term recurrence equation satisfied by a family of orthogonal polynomials, the Christoffel-Darboux-type bilinear generating function and their asymptotic expressions, we obtain quadrature formulas for integral transforms…
The explicit construction of direct and inverse Fourier's vector transform with discontinuous coefficients is presented. The technique of applying Fourier's vector transform with discontinuous coefficients for solving problems of…
In this paper we describe an intrinsically geometric way of producing magnetic fields on $\S^3$ and $\R^3$ for which the corresponding Dirac operators have a non-trivial kernel. In many cases we are able to compute the dimension of the…
In this paper we prove the Fourier restriction theorem for $p=2$ on Riemannian symmetric spaces of noncompact type with real rank one which extends the earlier result proved in \cite[Theorem 1.1]{KRS}. This result depends on the weak $L^2$…
We derive an explicit inversion algorithm for the spherical Radon transform in odd dimensions with partial radial data. We prove that the reconstruction of the unknown function can be reduced to solving ordinary differential equations,…
We present an explicit construction of the solution to the Dirichlet boundary value problem for the radial Schr\"odinger equation in the unit ball, with a complex-valued potential $V$ satisfying the condition $\int_0^1r|V(r)|dr<\infty$. The…
We present and demonstrate a method for optical homodyne tomography based on the inverse Radon transform. Different from the usual filtered back-projection algorithm, this method uses an appropriate polynomial series to expand the Wigner…
In this paper, estimates are proven for convolution kernels associated to multipliers from a reasonably general class of compactly supported two-dimensional functions constructed out of real-analytic functions. These estimates are both for…
We revisit Rahimi and Recht (2007)'s kernel random Fourier features (RFF) method through the lens of the PAC-Bayesian theory. While the primary goal of RFF is to approximate a kernel, we look at the Fourier transform as a prior distribution…
In this paper, we investigate the trigonometric Heckman-Opdam polynomials of type $A_1$. We establish connections with ultraspherical polynomials and derive an explicit expression for the associated Poisson kernel. Using the product…
This paper investigates the formulation and implementation of Bayesian inverse problems to learn input parameters of partial differential equations (PDEs) defined on manifolds. Specifically, we study the inverse problem of determining the…
The Laplace transform is a useful and powerful analytic tool with applications to several areas of applied mathematics, including differential equations, probability and statistics. Similarly to the inversion of the Fourier transform,…
A method for deforming C*-algebras is introduced, which applies to C*-algebras that can be described as the cross-sectional C*-algebra of a Fell bundle. Several well known examples of non-commutative algebras, usually obtained by deforming…
We revisit the Fourier transform of a Hankel function, of considerable importance in the theory of knife edge diffraction. Our approach is based directly upon the underlying Bessel equation, which admits manipulation into an alternate…
New high statistics data from the second generation of ultrarelativistic heavy-ion experiments open up new possibilities in terms of data analysis. To fully utilize the potential we propose to analyze the $m_\perp$-spectra of hadrons using…