  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.
