Theorems of Chernoff and Ingham for certain eigenfunction expansions

TL;DR

This paper establishes an uncertainty principle for specific eigenfunction expansions and applies it to derive analogues of Chernoff and Ingham theorems for various differential operators on symmetric spaces and Euclidean spaces.

Contribution

It introduces a new uncertainty principle for eigenfunction expansions and extends classical theorems to a broader class of operators and spaces.

Findings

Proved an uncertainty principle for eigenfunction expansions in weighted L^2 spaces.

Derived Chernoff and Ingham type theorems for Laplace-Beltrami and Hermite operators.

Extended classical harmonic analysis results to symmetric spaces and complex Euclidean spaces.

Abstract

We prove an uncertainty principle for certain eigenfunction expansions on $ L^2(\mathbb{R}^+,w(r)dr) $ and use it to prove analogues of theorems of Chernoff and Ingham for Laplace-Beltrami operators on compact symmetric spaces, special Hermite operator on $ \mathbb{C}^n $ and Hermite operator on $ \mathbb{R}^n.$