An Uncertainty Principle of Paley and Wiener on Euclidean Motion Group

Abstract

A classical result due to Paley and Wiener characterizes the existence of a nonzero function in L 2 (R), supported on a half line, in terms of the decay of its Fourier transform. In this paper we prove an analogue of this result for compactly supported continuous functions on the Euclidean motion group M (n). We also relate this result to a uniqueness property of solutions to the initial value problem for time-dependent Schrödinger equation on M (n).MSC 2010 : Primary 22E30; Secondary 43A80.

Uncertainty principles of Ingham and Paley–Wiener on semisimple Lie groups

“…We refer the reader to [13] and the references therein for results in this regard. These results were further generalized in the context of noncommutative groups in [7,29,1,26,2,3]. In this section we wish to relate the theorems of Ingham and Paley-Wiener to the above mentioned problem in the context of symmetric spaces.…”

“…To prove the first theorem we shall prove a lemma on entire functions on C n . The proof of this lemma for the one-variable case is given in [3]. Since the paper is yet to be accepted, we reproduce the proof here for the sake of completeness.…”

A Result of Paley and Wiener on Damek–ricci spaces

“…is a bijection. A weaker analogue of Theorem 1.1 has been proved in [4,Theorem 2.3] assuming that the function is compactly supported. We now state and prove an exact analogue of Theorem 1.1 for R d .…”

“…Paley and Wiener proved Theorem 1.2 using complex analytic techniques via a holomorphic extension of the Fourier transform in the upper half-plane. This complex analytic technique motivated us to prove an analogue of Theorem 1.1 for compactly supported smooth functions on the Euclidean motion group and connected, noncompact, semisimple Lie groups with finite center [4,5].…”

“…Since for a noncompact, semisimple Lie group G there is no natural way of defining functions supported on a half-space, in [4,5] we have worked under the assumption that f ∈ C ∞ c (G). However, the situation changes if we talk about functions defined on X = G/K, where K is a maximal compact subgroup of G. To explain this point let us consider the Iwasawa decomposition G = NAK where A = exp a with dim a = 1.…”