Notes on some papers of V. Komornik on vibrating membranes (Q804352)

This note collects several results pertaining to the completeness in \(L_ 2\)-spaces, in Sobolev spaces and in \(C_ k\)-spaces, \(k\in {\mathbb{N}}\), of sets of functions of the form \(x^ me^{2\pi i\lambda x}\), where \(\lambda\), m, are taken from various sub sets of \({\mathbb{R}}\) and \({\mathbb{N}}\), respectively. Sufficient conditions on (\(\lambda\),m) parameter sets are described in terms of their densities. Such questions are related to the work of \textit{V. Komornik} referred to in the title [e.g.: Bull. Sci. Math., II. Sér. 113, No.3, 289-308 (1989; Zbl 0719.42018); Proc. R. Soc. Edinb., Sect. A 111, No.1/2, 13-20 (1989; Zbl 0676.73029); with \textit{A. Haraux}, Pitman Res. Notes Mat. Ser. 166, 110-119 (1988; Zbl 0671.35054); J. Math. Anal. Appl. 122, 538-554 (1987; Zbl 0619.35065)]. As an application the 2-dimensional wave equation (equation of vibrating membrane) in the unit disk \(\Omega\) (with Dirichlet boundary condition) is considered. The set of N-tuples \((P_ 1,P_ 2,...,P_ N)\in \Omega^ N\) such that \(\{(u(\cdot,P_ 1)\), \(u(\cdot,P_ 2),...,u(\cdot,P_ N))|\) u is solution\(\}\) is dense in \([C_{\infty}({\mathbb{R}})]^ N\), is found to be itself dense in \(\Omega^ N\).