The Fourier-Walsh subsequence of partial sums
For a nonnegative integer \(n\) let us denote \(s(n) = \sum^ \infty_{i = 0} | \varepsilon_ i - \varepsilon_{i + 1} |\), where \(n = \sum^ \infty_{i = 0} \varepsilon_ i2^ i\), \(\varepsilon_ i \in \{0,1\}\). We recall that a Walsh function \(w_ n(x)\) is \(\prod^ k_{i = 0} (r_ i (x))^{\varepsilon_ i}\), where \(r_ i (x) = \text{sign} \sin 2^{i + 1} \pi x\). For a function \(f \in L[0,1)\) we denote the \((n- 1)\)-th sum of its Fourier-Walsh series by \(S_ n(f)\). There exists a Fourier-Walsh series that diverges at all dyadic irrational points [\textit{B. I. Golubov}, \textit{A. V. Efimov} and \textit{V. A. Skvortsov}: ``Walsh series and transformations. Theory and applications (1987; Zbl 0692.42009; English translation 1991; Zbl 0785.42010); Theorem 9.1.2]. On the other hand, it is known that if \(n_ j < n_{j + 1}\), \(\sup_ j s(n_ j) < \infty\), then \(S_{n_ j} (f) \to f\) a.e. L. A. Balashov has posed the following problem. Let a sequence \(\{n_ j\}\) be such that \(\sup_ j s(n_ j) = \infty\), which is equivalent to the unboundedness of the Lebesgue constants \(L_{n_ j}\). Does the existence of a function \(f \in L[0,1)\) with a sequence \(\{S_{n_ j} (f)\}\) divergent a.e. follow from the latter? The answer is negative.
scientific article; zbMATH DE number 687521