Best Approximation in the Supremum Norm by Analytic and Harmonic Functions

摘要

In this paper, the authors study the problem of finding, for a given bounded211u001emeasurable function f on a domain omega in R(sup n), a harmonic function on omega 211u001ethat best approximates f in the supermum norm, as well as (when n = 2) the 211u001ecorresponding problem of approximating a bounded measurable function on the 211u001eboundary of a plane domain (especially, the unit disk) by the boundary values of 211u001ebounded analytic functions in the interior has been studied very extensively, but 211u001ethe present problem (which, as we shall see, is quite different in character) has 211u001ereceived very little attention. There have been some studies, by Luecking (Lu1, 211u001eLu2), Hintzman (Hi1, Hi2) and Romanova (Ro1, Ro2), pertaining to approximation by 211u001eanalytic functions.