Muntz-Szasz type theorems for the density of the span of powers of functions

摘要

The aim of this paper is to establish density properties in L-P spaces of the span of powers of functions {psi(lambda): lambda is an element of Lambda}, Lambda subset of N in the spirit of the Miintz-Szasz Theorem. As density is almost never achieved, we further investigate the density of powers and a modulation of powers {psi(lambda), psi(lambda)e(i alpha t): lambda is an element of Lambda}. Finally, we establish a Muntz-Szasz Theorem for density of translates of powers of cosines {cos(lambda) (t - theta(1)), cos(lambda)(t - theta(2)): lambda is an element of Lambda}. Under some arithmetic restrictions on theta(1)-theta(2), we show that density is equivalent to a Muntz-Szasz condition on Lambda and we conjecture that those arithmetic restrictions are not needed. Some links are also established with the recently introduced concept of Heisenberg Uniqueness Pairs. (C) 2020 Elsevier Masson SAS. All rights reserved.