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
