Math. Nachr. 263–264, 3 – 35 (2004) / DOI 10.1002/mana.200310121 Generalized approximation spaces and applications Jose Maria Almira ∗1 and Uwe Luther ∗∗2 1 Departamento de Matematicas, Universidad de Ja´ en, E. U. P. Linares, C/ Alfonso X el sabio 28, 23700 Linares (Ja´ en), Spain 2 Fakult¨ at f¨ ur Mathematik, Technische Universit¨ at Chemnitz, 09107 Chemnitz, Germany Received 30 October 2001, revised 5 March 2002, accepted 10 June 2002 Published online 16 December 2003 Key words Approximation spaces, Besov spaces, extension of unbounded operators, representation theorem MSC (2000) 41A65 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. c  2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 1 Introduction One of the main tasks in approximation theory is the following problem: How good can complicated functions of a large space X be approximated by more simple functions (as polynomials, splines or rational functions) which belong to a certain subset A ⊂ X . For example, we may take X = L p 2π , the space of all 2π-periodic L p -functions on R, and A = T , the set of all trigonometric polynomials. Then, for any f ∈ L p 2π , we may define the sequence of errors of best approximation {E ∗ n (f )} n∈N , where E ∗ n (f ) = inf T ∈Tn ‖f - T ‖ Lp(0,2π) and T n is the set of all trigonometric polynomials of degree less than n. It is natural to hope that some rule of the kind “as smoother the function as smaller the best approximation errors” holds true and in fact it is one of the basic aims in approximation theory to find out the connection between smoothness and best approximation errors. For approximation by trigonometric polynomials it were Jackson and Bernstein who solved this problem proving the so called direct and inverse theorems. The inverse was more difficult and did not have an algebraic counterpart until the work of Timan, Nikolskii, Dzyadyk and others. From direct and inverse theorems it follows that certain classical function spaces can be viewed as special approximation spaces (see Section 3.1 for the definition of these spaces) and in our opinion this is one of the best mathematical expression of equivalences between the degree of smoothness of functions and the behaviour of their best approximation errors. For example, the space of all 2π-periodic functions on R, which are H¨ older continuous with exponent α ∈ (0, 1), is equal to the approximation space {f ∈ C 2π : {(n + 1) α E ∗ n (f )}∈ l ∞ }, where E ∗ n (f ) denotes the error of best approximation by trigonometric polynomials of degree less than n in the supremum norm on [0, 2π]. Other classical function spaces coresspond to approximation spaces A(X, S )= {f ∈ X : {E n (f )}∈ S} (E n (f ): error of best approximation by elements of a certain subset A n of X ) in which the sequence space S is of the form S = l q ( (n + 1) s-(1/q) ) , s> 0 (see Section 2 for some examples). The aim of the paper is to extend the well-known theory of such classical approximation spaces to approxima- tion spaces in which the sequence space S can be more or less arbitrary (in the sense that only a few properties are assumed to ensure that A(X, S ) is a quasi-normed space). This is not a new idea: Spaces of the type A(X, S ) are already studied by Brudnyi and Krugliak (see [8], Section 4.3.C). But they concentrate on interpolation prop- erties and only a few other results are proved directly without interpolation theory. For example, one can obtain reiteration and embedding theorems for approximation spaces via the coressponding theorems for interpolation spaces, but it may be hard to check the assumptions and the meaning of these theorems in case of non-classical S. ∗ e-mail: jmalmira@ujaen.es, Phone: +34 953 026503, Fax: +34 953 026508 ∗∗ Corresponding author: e-mail: uwe.luther@mathematik.tu-chemnitz.de, Phone: +49 371 5312159, Fax: +49 371 5312141 c  2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim