2 research outputs found
Approximation schemes satisfying Shapiro's Theorem
An approximation scheme is a family of homogeneous subsets of a
quasi-Banach space , such that , , and . Continuing the
line of research originating at a classical paper by S.N. Bernstein (in 1938),
we give several characterizations of the approximation schemes with the
property that, for every sequence , there exists
such that (in this case we
say that satisfies Shapiro's Theorem). If is a Banach space,
as above exists if and only if, for every sequence , there exists such that . We
give numerous examples of approximation schemes satisfying Shapiro's Theorem.Comment: 41 pages, Submitted to a Journal. A natural continuation of this
paper is also downloadable at Arxiv: See J. M. Almira and T. Oikhberg,
"Shapiro's theorem for Subspaces", at arXiv:1009.5535v
Characterization of approximation schemes satisfying Shapiro's Theorem
In this paper we characterize the approximation schemes that satisfy
Shapiro's theorem and we use this result for several classical approximation
processes. In particular, we study approximation of operators by finite rank
operators and n-term approximation for several dictionaries and norms.
Moreover, we compare our main theorem with a classical result by Yu. Brundyi
and we show two examples of approximation schemes that do not satisfy Shapiro's
theorem.Comment: This paper has been withdrawn by the author because a full revision
of it, with new and powerful contents, has been made with the help of another
author and now the paper has been transformed in another completely different
one which is common work with T. Oikhber