Metrical approximations of functions

In this paper, we analyze metrical approximations of functions $F :\Lambda times X \rightarrow Y$ by trigonometric polynomials and $\rho$-periodic type functions, where $\emptyset \neq \Lambda \subseteq {\mathbb R}^{n},$ $X$ and $Y$are complex Banach spaces, and $\rho$ 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 integro-differential equations and the partial differential equations.