Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 42, 2017, 809-836

SOME REMARKS ON ALMOST PERIODIC FUNCTIONS IN VIEW OF THE LEBESGUE MEASURE WITH APPLICATIONS TO LINEAR DIFFERENTIAL EQUATIONS

Dariusz Bugajewski and Adam Nawrocki

Adam Mickiewicz University, Faculty of Mathematics and Computer Science
ul. Umultowska 87, 61-614 Poznan, Poland; ddbb 'at' amu.edu.pl

Adam Mickiewicz University, Faculty of Mathematics and Computer Science
ul. Umultowska 87, 61-614 Poznan, Poland; adam.nawrocki 'at' amu.edu.pl

Abstract. In this paper we are going to investigate some properties of almost periodic functions in view of the Lebesgue measure with particular emphasis on their behavior under convolution. These considerations allow us to establish the main result concerning almost periodic in view of the Lebesgue measure solutions to linear differential equations of the first order. We also apply the theory of continued fractions to examine asymptotic behavior of a certain classical almost periodic function of that type. For that purpose we provide a new general method of calculation of certain type of limits.

2010 Mathematics Subject Classification: Primary 42A75, 42A85, 34A30, 26A03, 11A55.

Key words: Almost periodic function in view of the Lebesgue measure, best rational approximation, continued fraction, convolution, kth convergent, limit, linear differential equation, Stepanov almost periodic function.

Reference to this article: D. Bugajewski and A. Nawrocki: Some remarks on almost periodic functions in view of the Lebesgue measure with applications to linear differential equations. Ann. Acad. Sci. Fenn. Math. 42 (2017), 809-836.

Full document as PDF file

https://doi.org/10.5186/aasfm.2017.4250