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On strict inclusion relations between approximation and interpolation spaces. (English) Zbl 1222.41046

Summary: Approximation spaces, in their many presentations, are well known mathematical objects and many authors have studied them for long time. They were introduced by P. L. Butzer and K. Scherer [Approximationsprozesse und Interpolationsmethoden. Mannheim-Zürich: Bibliographisches Institut (1968; Zbl 0177.08501)] in 1968 and, independently, by Ju. A. Brudnyĭ and N. Ja. Kruglyak [A family of approximation spaces. Studies in the theory of functions of several real variables, No. 2, pp. 15–42, Yaroslav. Gos. Univ., Yaroslavl’ (1978)], and popularized by A. Pietsch [J. Approximation Theory 32, 115–134 (1981; Zbl 0489.47008)] in his seminal paper of 1981.

Pietsch was interested in the parallelism that exists between the theories of approximation spaces and interpolation spaces, so that he proved embedding, reiteration and representation results for approximation spaces.

In particular, embedding results are a natural part of the theory since its inception. The main goal of this paper is to prove that, for certain classes of approximation schemes $\left(X,{A}_{n}\right)$ and sequence spaces $S$, if ${S}_{1}\subset {S}_{2}\subset {c}_{0}$ (with strict inclusions) then the approximation space $A\left(X,{S}_{1},{A}_{n}\right)$ is properly contained into $A\left(X,{S}_{2},{A}_{n}\right)$. We also initiate a study of strict inclusions between interpolation spaces, for Petree’s real interpolation method.

##### MSC:
 41A65 Abstract approximation theory 41A25 Rate of convergence, degree of approximation 41A35 Approximation by operators (in particular, by integral operators) 41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities) 46B70 Interpolation between normed linear spaces 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
##### Keywords:
approximation space; real interpolation space; embedding