On boundedly divergent Walsh-Fourier series
Let \(f^*(x)\) represent the dyadic maximal function of an \(f\in L^ 1[0,1]\) and for any continuous increasing function \(\Phi: [0,\infty)\to [0,\infty)\) let \(H\Phi (H)\) denote the set of functions in \(L^ 1[0,1]\) which satisfy \(\int^{1}_{0}f^*(x)\Phi (f^*(x))dx<\infty.\) The author proves that if \(\Phi (t)=o(\log \log t)\) as \(t\to \infty\) then there is an \(F\in H\Phi (H)\) such that the Walsh-Fourier series of F diverges boundedly a.e. and diverges everywhere on \([0,1].\) The method of proof is elegant and direct, using Waterman's characterization of strongly multiplicative systems to estimate the Walsh-Kolmogorov polynomials and a stopping time like argument to decompose the interval \([0,1]\).
scientific article; zbMATH DE number 4205094