*(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 $(X,{A}_{n})$ 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:

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 |