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In this paper, we analyze metrical approximations of functions $F :\Lambdatimes X \rightarrow Y$ by trigonometric polynomials and $\rho$-periodic typefunctions, 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 Dossgeneralized approaches to metrical approximations. We clarify many structuralproperties of introduced spaces of functions and provide several applicationsof our theoretical results to the abstract Volterra integro-differentialequations and the partial differential equations.

Metrical approximations of functions