In the paper we generalize the theory of classical approximation spaces to a much wider class of spaces which are defined with the help of best approximation errors. We also give some applications. For example, we show that generalized approximation spaces can be used to find natural (in some sense) domains of definition of unbounded operators.

We show that generalized approximation spaces can be used to describe the relatively compact sets of Banach spaces. This leads to compactness and convergence criteria in the approximation spaces themselves. If these spaces can be described with the help of moduli of smoothness, then the criteria can be formulated in terms of the moduli. As applications we give a generalization of Bernstein's theorem about existence of elements with prescribed best approximation errors, compactness criteria for operators, a criterion for compactness in Sobolev type spaces, and a generalization of Simon's compactness criterion for subsets of L p -spaces of Banachspace-valued functions.

We study the finite dimensional spaces V which are invariant under the action of the finite differences operator ∆ m h . Concretely, we prove that if V is such an space, there exists a finite dimensional translation invariant space W such that V ⊆ W . In particular, all elements of V are exponential polynomials. Furthermore, V admits a decomposition V = P ⊕ E with P a space of polynomials and E a translation invariant space. As a consequence of this study, we prove a generalization of a famous result by P. Montel [7] which states that, if f : R → C is a continuous function satisfying ∆ m h 1 f (t) = ∆ m h 2 f (t) = 0 for all t ∈ R and certain h1, h2 ∈ R\{0} such that h1/h2 ∈ Q, then f (t) = a0+a1t+· · ·+am−1t m−1 for all t ∈ R and certain complex numbers a0, a1, · · · , am−1. We demonstrate, with quite different arguments, the same result not only for ordinary functions f (t) but also for complex valued distributions. Finally, we also consider in this paper the subspaces V which are ∆ h 1 h 2 ···hm -invariant for all h1, · · · , hm ∈ R.

An approximation scheme is a family of homogeneous subsets (A n ) of a quasi-Banach space X, such that A 1 A 2 .

In this paper we give a new proof of a classical result by Fréchet [M. Fréchet, Une définition fonctionnelle des polynomes, Nouv. Ann. 9 (4) (1909) 145-162]. Concretely, we prove that, if Δ k+1 h f = 0 and f is continuous at some point or bounded at some nonempty open set, then f ∈ P k . Moreover, as a consequence of the technique developed for our proof, it is possible to give a description of the closure of the graph for the solutions of the equation. Finally, we characterize some spaces of polynomials of several variables by the use of adequate generalizations of the forward differences operator Δ k+1 h .