Around Uncertainty Principles of Ingham-type on $\R^n$, $\T^n$ and Two Step Nilpotent Lie Groups

Abstract

Classical results due to Ingham and Paley-Wiener characterize the existence of nonzero functions supported on certain subsets of the real line in terms of the pointwise decay of the Fourier transforms. We view these results as uncertainty principles for Fourier transforms. We prove certain analogues of these uncertainty principles on the n-dimensional Euclidean space, the n-dimensional torus and connected, simply connected two step nilpotent Lie groups. We also use these results to show a unique continuation property of solutions to the initial value problem for time-dependent Schrödinger equations on the Euclidean space and a class of connected, simply connected two step nilpotent Lie groups.

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

“…All these results mentioned above are obtained only for the circle group and real line. Only very recently we have obtained the following analogues of the results of Paley-Wiener and Ingham in the context of n-dimensional Euclidean spaces (see Theorem 2.3 and Theorem 2.2 of [2] respectively). For f ∈ L 1 (R n ), we shall define its Fourier transform f by…”

“…We shall prove an analogue of Theorem 1.1 (b) for bi-K-invariant functions on a noncompact, complex semisimple Lie group using the explicit expression of the elementary spherical functions φ λ available in this case. First we shall recall the following result proved in [2] which is needed in our proof. Lemma 3.8.…”

“…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.…”