On a Problem of Hardy for Walsh-Fourier Series

A tight bound (on a logarithmic scale) for the order of growth almost everywhere of partial sums of Walsh- Fourier series has been reported. The currently best lower bound was derived by Konyagin and differs from by the root of a logarithm. The proof of Theorem 1 makes use of an important quadratic function generalizing the Paley function. The result cannot be substantially refined any more and, in a sense, is final.