On moments and symmetrical sequences

Abstract

In this article we consider questions related to the behavior of the moments Mmzj when the indices are restricted to specific subsequences of integers, such as the even or odd moments. If n≥2 we introduce the notion of symmetrical series of order n, showing that if zj is symmetrical then Mmzj=0 whenever n∤m; in particular, the odd moments of a symmetrical series of order 2 vanish. We prove that when zj∈lp for some p then several results characterizing the sequence from its moments hold. We show, in particular, that if Mmzj=0 whenever n∤m then zj is a rearrangement of a symmetrical series of order n. We then construct examples of sequences whose moments vanish with required density. Lastly, we construct counterexamples of several of the results valid in the lp case if we allow the moment series to be all conditionally convergent. We show that for each arbitrary sequence of real numbers μmm=0∞ there are real sequences ujj=0∞ such that ∑j=0∞uj2m+1=μmm≥0.