A note on the moments of sequences of complex numbers
aa r X i v : . [ m a t h . C V ] A p r A note on the moments of sequences of complex numbers
Maher Boudabra, Greg MarkowskyMonash UniversityApril 7, 2020
Abstract
We give a short proof that the limsup of the p -th root of the modulus of the p -th momentof a sequence of complex numbers is equal to the modulus of the maximum of the sequence.This strengthens known results, and provides an analog to a recent result concerning moments ofcomplex polynomials. Given a complex-valued function f defined on the interval [0; 1℄ , let the p -th moment of f be definedas Mp = R 10 f(x)pdx . In the recent paper [3], the following intriguing result was proved.
Theorem 1.
Suppose f is a polynomial with complex coefficients that is not identically 0. Then lim supp!1 j Mp j1=p> 0 . The authors also ventured the natural conjecture that perhaps lim supp!1 j Mp j1=p is equal to maxx2[0;1℄ j f(x) j . Note that this theorem and the conjecture may appear simple, given the ease withwhich they may be proved if f is assumed to be real-valued, but the analytic proof given in [3] is farfrom easy. Upon reflection, this is natural: the moments of f(x) = e2(cid:25)ix are all zero, and so any proofof this theorem must depend heavily upon properties of polynomials. An immediate consequence ofthis result is that the only polynomial with all moments 0 is the polynomial with all coefficients 0, aresult which had earlier been given an algebraic proof (which is also not simple), in [1].Motivated by this result, we have considered the analogous result for sequences of complex numbers.Similarly to above, given a sequence of complex numbers fzngn denote by Mp for positive integer p the sum P+1n=0 zpn . Our result is as follows.
Theorem 2.
Suppose a sequence fzngn is in `q for some q (cid:21) 1 ; that is, P+1n=0 jznjq < 1 . Then lim supp!1 j Mp j 1p = maxn j zn j :
An immediate consequence is the following.
Corollary 3.
Suppose a sequence fzngn is in `q for some q (cid:21) 1 , and Mp = 0 for all p . Then zn = 0 for all n . Before proving our the theorem, several remarks are in order. First, the corollary has already beenproven, using a rather technical argument involving C´es`aro summation, in [4]. In contrast, our proof ofTheorem 2 is very short, and makes use of generating functions and elementary complex analysis. Nextwe note that the condition that our sequence is in `q is necessary: in [2] a remarkable series of nonzerocomplex numbers is constructed, all of whose moments are 0. Naturally, this sequence is not in `q for any q . Finally, we remark that, in our opinion, our theorem lends credence to the aforementioned conjectureof M¨uger and Tuset on moments of polynomials, that lim supp!1 j Mp j1=p= maxx2[0;1℄ j f(x) j .Let us now prove Theorem 2. Required background for the proof can be found in virtually anycomplex analysis text (for instance, [5] or [6]). Let us begin by assuming that fzngn is in `1 , and bymultiplying by a constant we may also assume that maxn j zn j= 1 (we are assuming that zn 6= 0 forat least one n , otherwise the result is trivial). For w 2 D , define f(w) = 1Xn=1 znw1 (cid:0) znw : (0.1)1ince j1(cid:0)znwj is bounded uniformly away from on compact subsets of D , the fact that fzngn 2 `1 implies that the series in (0.1) converges locally uniformly, and f is therefore analytic on D . Byexpanding each term in the series into a geometric series and swapping the order of summation (whichis straightforward to justify using the locally uniform convergence), we find that the power seriesexpansion for f is simply the generating function for the sequence Mp , namely f(w) = 1Xp=1 Mpwp: (0.2)However, it is evident from (0.1) that limw!1=zn jf(w)j = 1 if jznj = 1 , and we have assumed thatthere is at least one such zn . This implies that f can not be extended to an analytic function on adisk centered at the origin of radius greater than one, and therefore that the radius of convergence ofthe power series in (0.2) is 1. In other words, lim supp!1 j Mp j 1p = 1 = maxn j zn j : Now we suppose that fzngn is not in `1 but is in `q for some positive integer q > 1 , and we stillassume maxn j zn j= 1 . Let vn = zqn , and note that Mp(v) = Mpq(z) . The prior argument applies to fvngn , since this sequence is in `1 , and so lim supp!1 j Mp(z) j 1p (cid:21) lim supp!1 j Mpq(z) j 1pq = (cid:16) lim supp!1 j Mp(v) j 1p (cid:17) 1q = 1: (0.3)On the other hand, the quantities j Mp(z) j are uniformly bounded for large enough p (since maxn j zn j= 1 and fzngn 2 `q ), so that lim supp!1 j Mp(z) j 1p (cid:20) 1 . The result follows. References [1] J.P. Francoise, F. Pakovich, Y. Yomdin, and W. Zhao. Moment vanishing problem and positivity:Some examples.
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