Completeness of trigonometric system with integer indices {e(inx); x is anelement of R}

Necessary and sufficient conditions are given which ensure the completeness of the trigonometric systems with integer indices; {e(inx); x is an elemen t of R}(n = -infinity)(infinity) or {e(inx); x is an element of R}(n = 1)(i nfinity) in L-alpha(mu, R), alpha greater than or equal to 1. If there exis ts a support A of the measure p which is a wandering set, that is, Lambda 2k pi, k = 0, +/-1, +/-2, ... are mutually disjoint for different k's, the n the linear span of our trigonometric system {e(inx); X is an element of R }(n = -infinity)(infinity) is dense in L-alpha(mu, R) alpha greater than or equal to 1. The converse statement is also true. (C) 2001 Academic Press