Decreasing rearranged Fourier series
In a previous paper, the author proved that there exists a real \(f\in L^2(T)\) whose Fourier sum, when taken in decreasing order of magnitude of the coefficients, diverges unboundedly almost everywhere. In the present paper this result is improved by taking \(f\) continuous. The proof follows the general lines of the former theorem but is substantially more intricate. The key new ingredients are a result of Kahane, establishing control over the size of the coefficients and the amplitude of certain trigonometric polynomials and a modified Olevskij lemma. In spite of the fact that the methods used in this paper are baroque (with the author's own word) this paper is remarkable, among others by the interesting comments covering the different steps of the proof.
scientific article; zbMATH DE number 1275884