Abstract
We investigate the possibility of approximating a function on a compact setK of the complex plane in such a way that the rate of approximation is almost optimal onK, and the rate inside the interior ofK is faster than on the whole ofK. We show that ifK has an external angle smaller than π at some point zo∈δK, then geometric convergence insideK is possible only for functions that are analytic at zo. We also consider the possibility of approximation rates of the form exp(−cn β), for approximation insideK, where β is related to the largest external angle ofK. It is also shown that no matter how slowly the sequence {γ n } tends to zero, there is aK and a Lip β, β<1, functionf such that approximation insideK cannot have order {γ n }.
Similar content being viewed by others
References
V. K. Dzjadyk (1959):On the problem of S. M. Nikolskii in the complex regions. Izv. Akad. Nauk. SSSR,23:697–736 (Russian).
V. V. Maimeskul (1992):Approximation of analytic functions by “near best” polynomial approximants. Ukrain. Mat. Zh.,44:208–214 (Russian).
E. B. Saff, V. Totik (1989):Behavior of polynomials of best uniform approximation. Trans. Amer. Math. Soc.,316:587–593.
N. A. Shirokov (1974):On uniform approximation of functions on closed sets with non-zero exterior angles. Izv. Akad. Nauk. Armyan. SSR,11:62–80 (Russian).
Author information
Authors and Affiliations
Additional information
Communicated by Dieter Gaier.
Rights and permissions
About this article
Cite this article
Shirokov, N.A., Totik, V. Polynomial approximation on the boundary and strictly inside. Constr. Approx 10, 145–152 (1994). https://doi.org/10.1007/BF01263060
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01263060