Luther, Uwe : A Note on Simultaneous Approximation

Author(s):

Luther, Uwe

Title:

A Note on Simultaneous Approximation

Electronic source:

application/pdf
application/postscript

Preprint series:

Technische Universität Chemnitz, Fakultät für Mathematik (Germany). Preprint 10, 2002

Mathematics Subject Classification:

41A65

[ Abstract approximation theory ]

Abstract:

The following result is well-known: If $f\in X$ ($X$: some normed function space) can be approximated of order $\|f-f_n\|_X\le c\inf_{g_n\in X_n}\|f-g_n\|_X=O(n^{-s-r})$ ($r,s>0$ fixed) by elements $f_1,f_2,\dotsc$ of certain subspaces $X_1\subseteq X_2\subseteq\dotsc$ for which the Bernstein inequalities $\|g_n\|_Y\le c\,n^r\|g_n\|_X,$ $g_n\in X_n$, hold true with some Banach space $Y\hookrightarrow X$ of smooth functions, then $\|f-f_n\|_Y= O(n^{-s})$. (Usually, $\|f\|_Y$ contains the norm of $f$ and some norm of $f^{(r)}$, so that $\|f-f_n\|_Y=O(n^{-s})$ means simultaneous approximation of $f$ and $f^{(r)}$ by $f_n$ and $f_n^{(r)}$, respectively.) We show that this result remains true if the order $O(a_n^{-1}n^{-r})$ is considered instead of $O(n^{-s-r})$, where $a_n$ is strictly increasing and converges to infinity faster than $n^\varepsilon$ (in a certain sense). We also present similar results in case $\sum (n^r\|f-f_n\|_X)^q(a_{n+1}^q-a_n^q)<\infty$ and in case of non-classical Bernstein inequalities, where $\{n^r\}$ is replaced by some other increasing sequence.

Keywords:

Simultaneous approximation, Best approximation errors

Language:

English

Publication time:

8 / 2002