UNIFORM APPROXIMATIONS BY QUASIPOLYNOMIALS WITH INTEGRAL COEFFICIENTS

In the light of these results, it seems natural to consider whether Muntz's theorem remains valid when all the coefficients of the approximating quasipolynomials are integers. This question was originally examined by the present author in [5]; in particular, it was shown there that, if the sequence of numbers {%~}~=0, tending to infinity, satisfies condition (i) and also certain assumptions about growth regularity, then any function / ~ C [0, I], for which f(O) and f(1) are integers, can be uniformly approximated in [0, i] by quasipolynomials with respect to the system of functions Jz~n} = with integral coefficients. The I n=O assumptions about growth regularity can be dropped when { n}~=1 is a subsequence of natural numbers (see [6]). But the question of whether a similar assertion holds in the general case still remains open.