Density theorems for almost periodic functions: a Hilbet space approach
Suppose \(T_ 1,T_ 2,...,T_ n\) be positive numbers and consider the vector space \(X=X_{T_ 1}+...+X_{T_ n}\), where \(X_{T_ i}\) denotes the vector space of all \(T_ i\)-periodic functions with zero mean, satisfying some smoothness condition. Several results have been known on the structure of X by \textit{A. Haraux} and the author [Rev. Mat. Iberoam. 1, No.4, 57-77 (1985; Zbl 0612.35077)] under the hypothesis \(T_ i/T_ j\) is irrational whenever \(i\neq j.\) The author proves that (i) X is dense in the space of all functions defined in an interval I and having the same smoothness property as X, if and only if the length of I is less than \(T_ 1+T_ 2+...+T_ n.\) (ii) In the limiting case where the length of I is equal to \(T_ 1+T_ 2+...+T_ n\), there exists a strictly positive function orthogonal to X. These results have direct applications to the study of a vibrating membrane. The main purpose of this paper is to remove the irrationality conditions, because this is useful for applications. The author solves the problem on the existence of a strictly positive orthogonal function in the limiting case. The present paper does not treat the applications of these results to oscillatory problems. Concerning this subject, one can see also papers by \textit{H. Brézis} [Bull. Am. Math. Soc., New Ser. 8, 409-426 (1983; Zbl 0515.35060)], and \textit{T. Cazenave} and \textit{A. Haraux} [C. R. Acad. Sci., Paris, Sér. I 298, 449-452 (1984; Zbl 0571.35074)].
scientific article; zbMATH DE number 4004757
scientific article