An uncertainty principle for eigenfunction expansions.
The aim of this paper is to present evidence that there are uncertainty principles associated to eigenfunction expansions. Several examples are given. If \(f\) is a radial function on a hyperboloid such that \(|f(x)| \leq C \exp(\alpha d (x,o)^2)\), \(o\) being a fixed reference point, and if the spherical transform \(\widetilde{f}\) of \(f\) satisfies \(|\widetilde{f} (\lambda)| \leq C \exp(-\beta |\lambda|^2)\), where \(\alpha \beta >1/4\), then \(f=0\). If \(f\) is a function on a compact connected analytic manifold \(M\), written as \(\sum f_k\), where the \(f_k\) are all eigenfunctions of a fixed analytic elliptic operator on \(M\), and if the \(f_k\) decay fast enough, then \(f\) must be analytic, and this cannot be localised in \(M\). Finally, a similar result is proved for Hermite expansions of functions on \(R^n\).
scientific article; zbMATH DE number 1153119