A note on convergence of Walsh-Fourier series of a function on a class \(V_{\alpha}\) (Q1091559)

For \(\alpha\geq 1\), let \(V_{\alpha}\) denote the set of periodic functions which are of bounded \(\alpha\)-th power variation in an interval of periodicity. \textit{N. Wiener} [Mass. J. Math. 3, 72-92 (1924)] proved that if \(f\in V_ 2\) then the (trigonometric) Fourier series of f is everywhere convergent. \textit{R. N. Siddiqi} [Can. Math. Bull. 20, 243-247 (1977; Zbl 0361.42006)] generalised this result by replacing \(V_ 2\) by \(V_{\alpha}\) for general \(\alpha\). The present paper considers the corresponding problem for Walsh-Fourier series. It is asserted that if \(f\in V_{\alpha}\) then its Walsh-Fourier series converges at a point \(x_ 0\) if and only if \(x_ 0\) is a point of continuity of f. However, the formula at the bottom of page 79 does not appear to be quite correct. As a simple example, consider the case \(m=2^ n\). With the notation of the paper, the formula given for \(S_ m(x_ 0)-f(x_ 0)\) reduces in this case to \(2^ n\int_{I^ 0_ n}\phi (x)dx=2^ n\int^{x_ 0+1/2^ n}_{x_ 0}f(x)dx-f(x_ 0).\) But the correct value is \(2^ n\int_{I_ n(x_ 0)}f(x)dx-f(x_ 0).\)