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

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.