Luther, U. : Representation, Interpolation, and Reiteration Theorems for Generalized Approximation Spaces

Author(s):

Luther, U.

Title:

Representation, Interpolation, and Reiteration Theorems for Generalized Approximation Spaces

Electronic source:

application/postscript

Preprint series:

Technische Universität Chemnitz, Fakultät für Mathematik (Germany). Preprint 12, 2001

Mathematics Subject Classification:

41A65			[ Abstract approximation theory ]
46B70			[ Interpolation between normed linear spaces ]
41A17			[ Inequalities in approximation ]

Abstract:

We show that the representation theorem for classical approxima tion spaces can be generalized to spaces $A(X,l^q(B))=\{f\in x:\{E_n(f)\}\in l^q{B)\}$ in which the weighted $l^q$-space $l^q(B)$ can be (more or less) arbitrary. We use this theorem to show that generalized approximation spaces can be viewed as real interpolation spaces (defined with K-functionals or main-part $K$-functionals) between couples of quasi-normed spaces which satisfy certain Jackson and Bernstein type inequalities. Especially, interpolation between an approximation space and the underlying quasi-normed space leads again to an approximation space. Together with a general reiteration theorem, which we also prove in the present paper, we obtain formulas for interpolation of two generalized approximation spaces.

Keywords:

Approximation spaces, Interpolation spaces, Representation and reiteration theorems, K-functionals and moduli of smoothness

Language:

English

Publication time:

3 / 2002