Fourier transforms related to a root system of rank 1

Inversion formula for the hypergeometric Fourier transform associated with a root system of type $BC$

We give the inversion formula and the Plancherel formula for the hypergeometric Fourier transform associated with a root system of type $BC$, when the multiplicity parameters are not necessarily nonnegative.

From Double Hecke algebra to Fourier transform

The paper contains a systematic theory of the one-dimensional Double Hecke algebra, including applications to the difference Fourier transform, Macdonald's polynomials, Gaussian sums at roots of unity, and Verlinde algebras. The main result is the classification of finite-dimensional representations for generic q and at roots of unity.

Clifford algebras, Fourier transforms and quantum mechanics

In this review, an overview is given of several recent generalizations of the Fourier transform, related to either the Lie algebra sl_2 or the Lie superalgebra osp(1|2). In the former case, one obtains scalar generalizations of the Fourier transform, including the fractional Fourier transform, the Dunkl transform, the radially deformed Fourier transform and the super Fourier transform. In the latter case, one has to use the framework of Clifford analysis and arrives at the Clifford-Fourier transform and the radially deformed hypercomplex Fourier transform. A detailed exposition of all these tr…

A Fourier transform for the quantum Toda lattice

We introduce an algebraic Fourier transform for the quantum Toda lattice.

单值混合 Hodge 模的 Fourier–Sato 变换

Fourier-Sato transform for monodromic mixed Hodge modules

本文研究单值混合 Hodge 模上的 Fourier–Sato 变换，并给出其构造方法。

The Abel, Fourier and Radon transforms on symmetric spaces

In this paper we prove a new inversion theorem and a refinement of an old support theorem for two Radon transforms on a symmetric space. Included are some new identities for the Abel transform and some results about the Fourier transform from a joint work with Rawat, Sengupta and Sitaram.