Some theorems on Fourier coefficients
where P is given by (1.1)? If one allows the coefficients Cn to be complex numbers of absolute value 1, an affirmative answer to the question is furnished by the partial sums of the series ZE inlogneinO; this example is due to Hardy and Littlewood [4, pp. 116-118]. A theorem of Salem and Zygmund [2, pp. 270, 278] shows, roughly speaking, that (N log N)12 is the most probable order of magnitude for ||P||X if C-n = ? 1. During the summer of 1958, Salem drew my attention to the problem stated in the first paragraph. It turns out that an affirmative answer can be given by a construction which uses nothing more sophisticated than the parallelogram law