An analogue of a theorem of Ferenc Lukács for double Walsh-Fourier series
A theorem of Ferenc Lukács states that if a periodic function \(f\) is integrable in the Lebesgue's sense and has discontinuity of first kind at some point \(\alpha\), then the \(n\)th partial sum of the conjugate series to its trigonometric Fourier series at \(\alpha\) divided by \(\log n\) converges to \(\pi^{-1}\{f(\alpha-0)-f(\alpha+0)\}\) as \(n\to\infty\). An analogue of this theorem for Walsh-Fourier series is proved by \textit{R. Riad} [Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Math. 30, 69-76 (1987; Zbl 0644.42025)]. The main aim of this paper is to extend the latter result from single to double Walsh-Fourier series. Given a number \(\alpha\in[0,1)\), consider its binary representation \(\alpha=\sum_{k=0}^{\infty}\alpha_k2^{-k-1}\). The \(n\)th variation \(V_n(\alpha)\) of \(\alpha\) is defined as \(V_n(\alpha):=|\alpha_0|+\sum_{k=0}^{n- }|\alpha_{k+1}-\alpha_k|\). Define the \(2^n\)th conjugate Walsh-Dirichlet kernel \(\tilde D_{2^n}^{\alpha} :=\sum_{k=0}^{n-1}(-1)^{r_k}D_{2^k}\), where \(r_k\) is the \(k\)th Rademacher function. The \(2^n\)th partial sum of the conjugate series is the dyadic convolution of the function \(f\) and \(\tilde D_{2^n}^{\alpha}\). Set \(\lambda_n(\alpha):=(\tilde S_{2^n}^{\alpha}\chi_{[0,\alpha)})(\alpha)\), where \(\chi_I\) denotes the characteristic function of the interval \(I\). The main theorem of the paper is as follows. \noindent Theorem. Let \(\alpha,\beta\in (0,1)\) be such that \(\lim _m(\alpha)=\lim V_n(\beta) =\infty\), and \(f\in L^1[0,1)^2\) such that \[ \int_0^1 |f(x,y)|dy , \int_0^1 |f(x,y)|dx \leq B \] for \(x,y\in (0,1)\), where \(B\) is some positive constant. If \(f\) has discontinuity of first kind at the point \((\alpha,\beta)\), then \[ \lim_{m,n}\frac{\tilde S_{2^m,2^n}^{\alpha,\beta}(\alpha,\beta)}{\lambda_m(\alpha)\lambda_n(\beta)} = f(\alpha+0, \beta+0)- f(\alpha-0, \beta+0)- f(\alpha+0, \beta-0)+ f(\alpha-0, \beta-0). \] The authors also consider functions of two variables which are of bounded variation over a rectangle in the sense of Hardy and Krause. Among others, they present a proof of the existence of the so-called sector limits of such functions at each point.
bounded variation over a rectangle in the sense of Hardy and Krause
sector limits of a function of two variables
scientific article; zbMATH DE number 1870473