Density theorems for almost periodic functions: a Hilbet space approach
type: Item
instance of: scholarly article
title: Density theorems for almost periodic functions: a Hilbet space approach
zbMATH Open document ID: 0619.35065
DOI: 10.1016/0022-247X(87)90283-6
published in: Journal of Mathematical Analysis and Applications
publication date: 1987-01-01T00:00:00Z
review text: 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)].
Mathematics Subject Classification ID: 35L05
Mathematics Subject Classification ID: 35B15
Mathematics Subject Classification ID: 35R20
Mathematics Subject Classification ID: 35B05
zbMATH DE Number: 4004757
zbMATH Keywords: density theorems
zbMATH Keywords: almost periodic function
zbMATH Keywords: orthogonal complement
zbMATH Keywords: polynomials
zbMATH Keywords: positive measure
zbMATH Keywords: periodic functions
zbMATH Keywords: zero mean
zbMATH Keywords: smoothness property
zbMATH Keywords: vibrating membrane
author: V. Komornik
reviewed by: Michael Th. Rassias
MaRDI profile type: Publication
cites work: Periodic solutions of nonlinear vibrating strings and duality principles
cites work: Q3688803
cites work: Oscillatory Phenomena Associated to Semilinear Wave Equations in One Spatial Dimension
cites work: Q3708397
cites work: Oscillations of anharmonic Fourier series and the wave equation
cites work: Q3736170
description: scientific article; zbMATH DE number 4004757