2013/02/12 by Schmidt, Kai-Uwe
#11B83 (Primary) #42A05 #94A55 (Secondary) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics
paper · doi:10.48550/arxiv.1302.2766
Littlewood raised the question of how slowly ||fn||44-||fn||24 (where ||.||r denotes the Lr norm on the unit circle) can grow for a sequence of polynomials fn with unimodular coefficients and increasing degree. The results of this paper are the following. For gn(z)=∑k=0n-1eπik2/n zk the limit of (||gn||44-||gn||24)/||gn||23 is 2/π, which resolves a mystery due to Littlewood. This is however not the best answer to Littlewood's question: for the polynomials hn(z)=∑j=0n-1∑k=0n-1 e2πijk/n znj+k the limit of (||hn||44-||hn||24)/||hn||23 is shown to be 4/π2. No sequence of polynomials with unimodular coefficients is known that gives a better answer to Littlewood's question. It is an open question as to whether such a sequence of polynomials exists.