caa a22 4500 475743792 CHVBK 20180406123507.0 cr unu---uuuuu 170329e20000401xx s 000 0 eng 10.1007/s003659910008 doi (NATIONALLICENCE)springer-10.1007/s003659910008 Polynomial Approximation on Convex Subsets of R n [Elektronische Daten] [Y. A. Brudnyi, N. J. Kalton] Abstract. : Let K be a closed bounded convex subset of R n ; then by a result of the first author, which extends a classical theorem of Whitney there is a constant w m (K) so that for every continuous function f on K there is a polynomial ϕ of degree at most m-1 so that |f(x)-ϕ(x)|≤ w_m(K) sup _{x,x+mh∈ K} |Δ_h^m(f;x)|. The aim of this paper is to study the constant w m (K) in terms of the dimension n and the geometry of K . For example, we show that w 2 (K)≤ (1/2) [ log 2 n]+5/4 and that for suitable K this bound is almost attained. We place special emphasis on the case when K is symmetric and so can be identified as the unit ball of finite-dimensional Banach space; then there are connections between the behavior of w m (K) and the geometry (particularly the Rademacher type) of the underlying Banach space. It is shown, for example, that if K is an ellipsoid then w 2 (K) is bounded, independent of dimension, and w 3 (K)\sim log n . We also give estimates for w 2 and w 3 for the unit ball of the spaces l p n where 1≤ p≤∈fty. Springer-Verlag New York Inc., 2000 Key words. Polynomial approximation, Whitney constant, Banach space. AMS Classification. 41A10 nationallicence Brudnyi Y. A. Department of Mathematics, Technion, 32000, Haifa, Israel aut Kalton N. J. Department of Mathematics, University of Missouri-Columbia, MO 65211, Columbia, USA aut https://doi.org/10.1007/s003659910008 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s003659910008 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Brudnyi Y. A. Department of Mathematics, Technion, 32000, Haifa, Israel aut NATIONALLICENCE 700 1- Kalton N. J. Department of Mathematics, University of Missouri-Columbia, MO 65211, Columbia, USA aut Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer