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Zbl 1222.41046
Almira, Jose Mar{\'\i}a
On strict inclusion relations between approximation and interpolation spaces.
(English)
[J] Banach J. Math. Anal. 5, No. 2, 93-105, electronic only (2011). ISSN 1735-8787/e

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 {\it P. L. Butzer} and {\it K. Scherer} [Approximationsprozesse und Interpolationsmethoden. Mannheim-Zürich: Bibliographisches Institut (1968; Zbl 0177.08501)] in 1968 and, independently, by {\it Ju. A. Brudny\u\i} and {\it 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 {\it 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 $(X,A_n)$ and sequence spaces $S$, if $S_1\subset S_2\subset c_0$ (with strict inclusions) then the approximation space $A(X, S_1,A_n)$ is properly contained into $A(X,S_2,A_n)$. We also initiate a study of strict inclusions between interpolation spaces, for Petree's real interpolation method.
MSC 2000:
*41A65 Abstract approximation theory
41A25 Degree of approximation, etc.
41A35 Approximation by operators
41A17 Inequalities in approximation
46B70 Interpolation between normed linear spaces
46E35 Sobolev spaces and generalizations

Keywords: approximation space; real interpolation space; embedding

Citations: Zbl 0177.08501; Zbl 0489.47008

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