Metrical approximations of functions

Abstract

In this paper, we analyze metrical approximations of functions F : Λ × X → Y by trigonometric polynomials and ρ-periodic type functions, where ∅ = Λ ⊆ R n , X and Y are complex Banach spaces, and ρ is a general binary relation on Y . Besides the classical concept, we analyze Stepanov, Weyl, Besicovitch and Doss generalized approaches to metrical approximations. We clarify many structural properties of introduced spaces of functions and provide several applications of our theoretical results to the abstract Volterra integrodifferential equations and the partial differential equations.2010 Mathematics Subject Classification. 42A75, 43A60, 47D99. Key words and phrases. Metrical approximations of functions by trigonometric polynomials, metrical approximations of functions by ρ-periodic type functions, abstract Volterra integrodifferential equations.

Generalized ρ-Almost Periodic Sequences and Applications

“…Let us finally mention some conclusions and final remarks about the introduced notion, some topics not considered here and some perspectives for further expansion of the theory. Before proceeding any further, we would like to emphasize that many other classes of generalized ρ-almost periodic type sequences are introduced and analyzed in our recent research studies [22][23][24][25]48,49]. For example, suppose that F : I × X → Y and (1) holds.…”