Related papers: New Inversion Formulas for the Horospherical Trans…
This is a brief overview of the index hypergeometric transform (other terms for this integral operator are: Olevskii transform, Jacobi transform, generalized Mehler--Fock transform). We discuss applications of this transform to special…
We establish shape holomorphy results for general weakly- and hyper-singular boundary integral operators arising from second-order partial differential equations in unbounded two-dimensional domains with multiple finite-length open arcs.…
Laplacian eigenmodes on homogeneous Clifford--Klein factors of the three--sphere are obtained as pullbacks of harmonics on the orbifolded two--sphere using the Hopf map. A method of obtaining these polyhedral, or crystal, harmonics using…
Using hyperbolic form convolution with doubly isometry-invariant kernels, the explicit expression of the inverse of the de Rham laplacian acting on m-forms in the Poincar\'{e} space is found. Also, by means of some estimates for hyperbolic…
In many applications data are measured or defined on a spherical manifold; spherical harmonic transforms are then required to access the frequency content of the data. We derive algorithms to perform forward and inverse spin spherical…
The generalized spherical Radon transform associates the mean values over spherical tori to a function $f$ defined on $\mathbb{S}^3 \subset \mathbb{H}$, where the elements of $\mathbb{S}^3$ are considered as quaternions representing…
An important class of fractional differential and integral operators is given by the theory of fractional calculus with respect to functions, sometimes called $\Psi$-fractional calculus. The operational calculus approach has proved useful…
In this work we extend consideration of the polynomial Carleson operator to the setting of a Radon transform acting along the paraboloid in $\mathbb{R}^{n+1}$ for $n \geq 2$. Inspired by work of Stein and Wainger on the original polynomial…
We study several fundamental harmonic analysis operators in the multi-dimensional context of the Dunkl harmonic oscillator and the underlying group of reflections isomorphic to $\mathbb{Z}_2^d$. Noteworthy, we admit negative values of the…
We present an approach to sums of random Hermitian matrices via the theory of spherical functions for the Gelfand pair $(\mathrm{U}(n) \ltimes \mathrm{Herm}(n), \mathrm{U}(n))$. It is inspired by a similar approach of Kieburg and K\"osters…
We present a rational version of the classical Landen transformation for elliptic integrals. This is employed to obtain explicit closed-form expressions for a large class of integrals of even rational functions and to develop an algorithm…
We introduce a Bargmann transform on the space of hyperplanes by applying the Plancherel formula of the Radon transform to the definition of the Bargmann transform on the Euclidean space. Some basic facts on microlocal analysis are also…
In this article we present a review of the Radon transform and the instability of the tomographic reconstruction process. We show some new mathematical results in tomography obtained by a variational formulation of the reconstruction…
We prove end point estimate for Radon transform of radial functions on affine Grasamannian and real hyperbolic space. We also discuss analogs of these results on the sphere.
A method for the semiclassical quantization of chaotic maps is proposed, which is based on harmonic inversion. The power of the technique is demonstrated for the baker's map as a prototype example of a chaotic map.
We introduce a new family of invariant differential operators associated with $\lambda$-cosine and Funk-Radon transforms on Stiefel and Grassmann manifolds. These operators reduce the order of the $\lambda$-cosine transforms and yield new…
We study a generalized spherical means operator, viz. generalized spherical mean Radon transform, acting on radial functions. As the main results, we find conditions for the associated maximal operator and its local variant to be bounded on…
We introduce a dimensional splitting method based on the intertwining property of the Radon transform, with a particular focus on its applications related to hyperbolic partial differential equations (PDEs). This dimensional splitting has…
In this paper extensions of the classical Fourier, fractional Fourier and Radon transforms to superspace are studied. Previously, a Fourier transform in superspace was already studied, but with a different kernel. In this work, the…
We develop isometry and inversion formulas for the Segal--Bargmann transform on odd-dimensional hyperbolic spaces that are as parallel as possible to the dual case of odd-dimensional spheres.