Related papers: Non-Euclidean Fourier inversion on super-hyperboli…

We have in [1] proposed a definition of cusp forms on semisimple symmetric spaces $G/H$, involving the notion of a Radon transform and a related Abel transform. For the real non-Riemannian hyperbolic spaces, we showed that there exists an…

We define an involution on the space of elliptic unipotent Langlands parameters of a reductive $p$-adic group $G$ and verify that when $G$ is split adjoint exceptional, the composition of this involution with the hyperspecial parahoric…

The Harish-Chandra--Howe local character expansion expresses the characters of reductive, $p$-adic groups in terms of Fourier transforms of nilpotent orbital integrals on their Lie algebras, and Murnaghan--Kirillov theory expresses many…

We consider the horospherical transform and its inversion in 3 examples of hyperboloids. We want to illustrate via these examples the fact that the horospherical inversion formulas can be directly extracted from the classical Radon…

Braverman and Kazhdan proposed a conjecture, later refined by Ng\^o and broadened to the framework of spherical varieties by Sakellaridis, that asserts that affine spherical varieties admit Schwartz spaces, Fourier transforms, and Poisson…

We investigate lower asymptotic bounds of number variances for invariant locally square-integrable random measures on Euclidean and real hyperbolic spaces. In the Euclidean case we show that there are subsequences of radii for which the…

This paper presents recent results obtained by the authors (partly in collaboration with A. Dabrowska) concerning expansions of zonal functions on Euclidean spheres into spherical harmonics and some applications of such expansions for…

We establish an entanglement principle for fractional powers of the Laplace-Beltrami operator on hyperbolic space $\mathbb H^n$, $n\ge 2$. More precisely, we prove that if finitely many distinct noninteger powers of $-\Delta_{\mathbb H^n}$,…

We generalize the notion of harmonic conjugate functions and Hilbert transforms to higher dimensional euclidean spaces, in the setting of differential forms and the Hodge-Dirac system. These conjugate functions are in general far from being…

Let G be a reductive connected p-adic group. With help of the Fourier inversion formula used in [Une formule de Plancherel pour l'algebre de Hecke d'un groupe reductif p-adique - V. Heiermann, Comm. Math. Helv. 76, 388-415, 2001] we give a…

Let $X$ be a complete, simply connected harmonic manifold of purely exponential volume growth. This class contains all non-flat harmonic manifolds of non-positive curvature and, in particular all known examples of harmonic manifolds except…

Any even function defined on 2-sphere is reconstructed from its integrals over big circles by means of the classical Funk formula. For the non-geodesic Funk transform on the sphere of arbitrary dimension, there is the explicit inversion…

We prove an inversion theorem for the Fourier transform defined for normal functions, in the case when such functions are of moderate decrease, and in dimensions 2 and 3. This improves on Carleson's general almost everywhere convergence…

The spherical Fourier transform on a harmonic Hadamard manifold $(X^n, g)$ of positive volume entropy is studied. If $(X^n, g)$ is of hypergeometric type, namely spherical functions of $X$ are represented by the Gauss hypergeometric…

Integral transforms are invaluable mathematical tools to map functions into spaces where they are easier to characterize. We introduce the hyperdimensional transform as a new kind of integral transform. It converts square-integrable…

Let $A$ be an expansive dilation on $\mathbb{R}^n$, and $p(\cdot):\mathbb{R}^n\rightarrow(0,\,\infty)$ be a variable exponent function satisfying the globally log-H\"{o}lder continuous condition. Let $\mathcal{H}^{p(\cdot)}_A({\mathbb…

The pseudo-polar Fourier transform is a specialized non-equally spaced Fourier transform, which evaluates the Fourier transform on a near-polar grid, known as the pseudo-polar grid. The advantage of the pseudo-polar grid over other…

In this paper, we generalize the weighted Fourier transform with respect to a function, originally proposed for the one-dimensional case in \cite{Dorrego}, to the $n$-dimensional Euclidean space $\mathbb{R}^{n}$. We develop a comprehensive…

Let $X$ be a complete, simply connected harmonic manifold with sectional curvatures $K$ satisfying $K \leq -1$, and let $\partial X$ denote the boundary at infinity of $X$. Let $h > 0$ denote the mean curvature of horospheres in $X$, and…

We show how the Fourier transform for distributional sections of vector bundles over symmetric spaces of non-compact type $G/K$ can be used for questions of solvability of systems of invariant differential equations in analogy to…