Random Rearrangements of Fourier Coefficients

Abstract

In [2] and [3] Hardy and Littlewood gave (1) necessary and sufficient conditions that numbers on, -0 should be, for every variation of their arguments and arrangement, the Fourier coefficients of a function in LP (2 < p < c), and (2) necessary and sufficient conditions that such numbers should be, for some variation of their arguments and arrangement, the Fourier coefficients of a function in LP (1 < p < 2). Results of Paley and Zygmund [6] have led to a characterization of those numbers n0 which, for every variation in their arguments, are the Fourier coefficients of a function in LP (1 ? p < oc). We ask the following question, which was inspired by these results: Which numbers on0 are, for every variation of their arrangement, the Fourier coefficients of a function in LP? Of course it is necessary to make precise the phrase almost every variation of their arrangement. Following Garsia [1] we consider a probability space X of local permutations of {1, 2, ... }. For k = 0, 1, ... let S(2k) be the symmetric group on the set {2k, 2k + 1, ** *, 2k~l1}. To each Uk e S(2k) we assign the probability 1/2k!, and we let X be the product probability space H0 S(2 k) For a = (0, al ... )eX and a Fourier series of the form E l it eino we define