We investigate the possibility of approximating a function on a compact set K of the complex plane in such a way that the rate of approximation is almost optimal on K, and the rate inside the interior of K is faster than on the whole of K. We show that if K has an external angle smaller than π at some point zo∈δK, then geometric convergence inside K is possible only for functions that are analytic at zo. We also consider the possibility of approximation rates of the form exp(-cnβ), for approximation inside K, where β is related to the largest external angle of K. It is also shown that no matter how slowly the sequence {γn} tends to zero, there is a K and a Lip β, β<1, function f such that approximation inside K cannot have order {γn}.
Original language | English |
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Pages (from-to) | 145-152 |
Number of pages | 8 |
Journal | Constructive Approximation |
Volume | 10 |
Issue number | 2 |
DOIs | |
State | Published - Jun 1994 |
Externally published | Yes |
ID: 86661982