On everywhere divergence of the strong Φ-means of Walsh–Fourier series

Almost everywhere strong exponential summability of Fourier series in Walsh and trigonometric systems was established by Rodin in 1990. We prove, that if the growth of a function Φ(t):[0,∞)→[0,∞)Φ(t):[0,∞)→[0,∞) is bigger than the exponent, then the strong Φ-summability of a Walsh–Fourier series can fail everywhere. The analogous theorem for trigonometric system was proved before by one of the authors of this paper.