Irreducible Tensor Operators and the Wigner-Eckart Theorem for Finite Magnetic Groups

The transformation properties of irreducible tensor operators and the applicability of the Wigner-Eckart theorem to finite magnetic groups have been studied.Comment: 9 pages, 0 figure

Function with its Fourier transform supported on annulus and eigenfunction of Laplacian

We explore the possibilities of reaching the characterization of eigenfunction of Laplacian as a degenerate case of the inverse Paley-Wiener theorem (characterizing functions whose Fourier transform is supported on a compact annulus) for the Riemannian symmetric spaces of noncompact type. Most distinguished prototypes of these spaces are the hyperbolic spaces. The statement and the proof of the main result work mutatis-mutandis for a number of spaces including Euclidean spaces and Damek-Ricci spaces.Comment: 24 page

On the Schwartz space isomorphism theorem for rank one symmetric space

In this paper we give a simpler proof of the $L^p$-Schwartz space isomorphism $(0< p\leq 2)$ under the Fourier transform for the class of functions of left $\delta$-type on a Riemannian symmetric space of rank one. Our treatment rests on Anker's \cite{A} proof of the corresponding result in the case of left $K$-invariant functions on $X$. Thus we give a proof which relies only on the Paley--Wiener theorem.Comment: 16 page

Beurling's Theorem and Lp−LqL^p-L^q Morgan's Theorem for Step Two Nilpotent Lie Groups

We prove Beurling's theorem and $L^p-L^q$ Morgan's theorem for step two nilpotent Lie groupsComment: 20 page

Beurling's Theorem and characterization of heat kernel for Riemannian Symmetric spaces of noncompact type

We prove Beurling's theorem for rank 1 Riemmanian symmetric spaces and relate it to the characterization of the heat kernel of the symmetric space

Asymptotic mean value property for eigenfunctions of the Laplace-Beltrami operator on Damek-Ricci spaces

Let $S$ be a Damek-Ricci space equipped with the Laplace-Beltrami operator $\Delta$. In this paper we characterize all eigenfunctions of $\Delta$ through sphere, ball and shell averages as the radius (of sphere, ball or shell) tends to infinity

Beurling's Theorem for SL(2,R)SL(2,\R)

We prove Beurling's theorem for the full group $SL(2,\R)$. This is the {\em master theorem} in the quantitative uncertainty principle as all the other theorems of this genre follow from it

Chaotic behaviour of the Fourier multipliers on Riemannian symmetric spaces of noncompact type

Let $X$ be a Riemannian symmetric space of noncompact type and $T$ be a linear translation-invariant operator which is bounded on $L^p(X)$. We shall show that if $T$ is not a constant multiple of identity then there exist complex constants $z$ such that $zT$ is chaotic on $L^p(X)$ when $p$ is in the sharp range $2<p<\infty$. This vastly generalizes the result that dynamics of the (perturbed) heat semigroup is chaotic on $X$ proved in [15, 17].Comment: 17 page

Cowling-Price theorem and characterization of heat kernel on symmetric spaces

We extend the uncertainty principle, the Cowling--Price theorem, on non-compact Riemannian symmetric spaces $X$. We establish a characterization of the heat kernel of the Laplace--Beltrami operator on $X$ from integral estimates of the Cowling--Price type.Comment: 22 pages, no figures, no table

A note on growth of Fourier transforms and Moduli of continuity on Damek Ricci spaces

We obtain results related to boundedness of the growth of Fourier transform by the modulus of continuity on Damek-Ricci spaces. For noncompact riemannian symmetric spaces of rank one, analogues of all the results follow the same way