An approach proposed by the author for representing spaces of local functions (s.l.f.) by means of an abstract function with values in the dual space is developed; thanks to this generalization of s.l.f. to the case of a nonuniform net and manifolds are obtained. Moreover, some weighted estimates of approximation by regular s.l.f. are established, and simple proofs are given of theorems of approximation by regular s.l.f. in the spaces of S. L. Sobolev and O. V. Besov.