Around theorems of Ingham-type regarding decay of Fourier transform on Rn, Tn

An uncertainty principle for spectral projections on rank one symmetric spaces of noncompact type

“…As remarked in [3] (see Remark 3.6), if the measure is even, then even polynomials are dense in L p e (ℝ, d ) , the subspace of even functions in L p (ℝ, d ). We will make use of this observation in the following proof.…”

“…Using a different approach, recently this result has been proved in [7]. Moreover, we remark that putting = 0 in the above result we can get a version of Ingham's theorem for the Fourier transform on ℝ n which was proved in [3] using a several variable version of the classical Denjoy-Carleman theorem for the quasi-analytic functions. Consequently, our result in the paper provides a new and simple proof of Ingham type uncertainty principle for the Fourier transform on ℝ n .…”

“…We will make use of this observation in the following proof. We remark that the proof given below is already present in [3] but for the sake of convenience of the reader we reproduce it here.…”

“…Ingham's theorem has received considerable attention in recent years and analogues have been proved for Fourier series [3] and Fourier transforms on symmetric spaces [4] and nilpotent Lie groups [1,3]. In this note we would like to recast some of the results as uncertainty principles for generalized spectral projections associated to Laplace-Beltrami operators on Riemannian symmetric spaces.…”

Theorems of Chernoff and Ingham for certain eigenfunction expansions

“…In the one dimensional case, this problem has been addressed by Ingham [10], Levinson [14], Paley and Wiener [15,16] and their results are in terms of non integrability of ψ(t) t −2 over [1, ∞). Recently this problem has received considerable attention and several versions have been proved in the contexts of R n , nilpotent Lie groups and compact and non-compact Riemannian symmetric spaces, see the works [2], [3], [4], [5] of Bhowmik, Pusti, Ray and Sen in various combinations of authorship. Recently, in a joint work with Bagchi and Sarkar [1] we have proved an analogue of Ingham's theorem for the operator valued Fourier transform on the Heisenberg group.…”