A Perturbation Theorem for Complete Sets of Complex Exponentials

The purpose of this note is to show that the completeness of a set of complex exponentials {e'x"'} in L2(—7r,w) is preserved whenever the \n are subjected to a suitable "lifting". There is an extensive literature on the completeness of sets of complex exponentials {e'x"') (see, for example, [l]-[8], and the references therein). In this note, we show that completeness is preserved in L?(—77,77) whenever the \n are subjected to a suitable "lifting". Theorem. Let {Xn} and {pn) be two sequences of points lying in a fixed horizontal strip and suppose that Re Xn = Re p„. // {eiX"!) is complete in L (—tt, tt), then so too is (e* **»'}. Proof. By making a suitable translation, we may assume that X„pn ¥= 0. Suppose that the set {e'^"') is not complete in L2(—tr,tt). Then there exists a function/0 in L2(—tt,tt) not equivalent to zero such that P Ut)e^'dt = 0 («= 1,2,...). J — 77 Let us denote by H the Paley-Wiener space of entire functions F of exponential type tt for which 11*11 = {/_! m*)i2rfx] < 00. If we set *o00 = f fo(t)ei2'dt, j—tt then F0 belongs to H, is not identically zero, and F0(pn) = 0 for each pn. We may suppose in addition that F0(0) = 1. This is clear if F0(0) ¥= 0, while if F0 has a zero of order m at the origin, then dividing F0 by a suitable multiple of zm produces the desired function. Let Fn(z) = F0{z) f[ P^T* (fl-1,2,...)k=x z pk \k Received by the editors August 6, 1975. AMS (MOS) subject classifications (1970). Primary 42A64; Secondary 46E30.