Asymptotics of Kolmogorov diameters for some classes of harmonic functionson spheroids

Let Gamma(D)(K) be the unit ball of the space of all bounded harmonic funct ions ina domain D in R-3, considered as a compact subset of the Banach spac e C(K), where K is a compact subset of D. The old problem about the exact a symptotics for Kolmogorov diameters (widths) of this set, ln d(k)(Gamma(D)(K)) similar to -tau k(1/2) , k --> infinity, is solved positively in the case when K and D are closed and open confocal spheroids, respectively (i.e., prolate or oblate ellipsoids of revolution). Using some special asymptotic formulas for the associated Legendre functio ns P-n(m)(cosh sigma) as n --> infinity and m/n --> gamma is an element of[ 0, 1] (considered earlier by the second author), we show that the constant tau is some averaged characteristic of the pair of spheroids, expressed by means of a certain function of the variable gamma, which appears within tho se asymptotics. Unlike the corresponding problem for analytic functions, qu ite well investigated, the harmonic functions case has been studied, up to now, only in the case of concentric balls. (C) 2000 Academic Press.