Joint Eigenfunctions On The Heisenberg Group And Support Theorems On Rn

by Amit Samanta

Institution: Indian Institute of Science
Year: 2012
Keywords: Harmonic Analysis; Hecke-Bochner Identity; Heisenberg Group; Convolution Equations; Eigenfunctions; K-Spherical Functions; Differential Operators; Weyl Transform; Recursion Theory; Mathematics
Record ID: 1184928
Full text PDF: http://hdl.handle.net/2005/2289


This work is concerned with two different problems in harmonic analysis, one on the Heisenberg group and other on Rn, as described in the following two paragraphs respectively. Let Hn be the (2n + 1)-dimensional Heisenberg group, and let K be a compact subgroup of U(n), such that (K, Hn) is a Gelfand pair. Also assume that the K-action on Cn is polar. We prove a Hecke-Bochner identity associated to the Gelfand pair (K, Hn). For the special case K = U(n), this was proved by Geller, giving a formula for the Weyl transform of a function f of the type f = Pg, where g is a radial function, and P a bigraded solid U(n)-harmonic polynomial. Using our general Hecke-Bochner identity we also characterize (under some conditions) joint eigenfunctions of all differential operators on Hn that are invariant under the action of K and the left action of Hn . We consider convolution equations of the type f * T = g, where f, g ε Lp(Rn) and T is a compactly supported distribution. Under natural assumptions on the zero set of the Fourier transform of T , we show that f is compactly supported, provided g is.