Abstract
This article concerns: 1) theorems on the spectra of operators formed from cosine operator-valued functions and representations; 2) inequalities (of Bernstein type) connecting the norms of operators with their spectral radii; 3) applications to second-order differential equations; 4) generalizations of the known spectral criterion of Loomis for almost periodicity, and an application to the investigation of almost periodicity of cosine operator-valued functions, representations, and solutions of functional equations; and 5) linear methods for summation of Fourier series in eigenfunctions of a linear operator generating a bounded (one-parameter) cosine operator-valued function.Bibliography: 41 titles.
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