Double Walsh series with coefficients of bounded variation of higher order

Let $D_{j}^{k}(x)$ denote the Cesàro sums of order $k$ of the Walsh functions. The estimates of $D_{j}^{k}(x)$ given by Fine back in 1949 are extended to the case $k>2$. As a corollary, the following properties are established for the rectangular partial sums of those double Walsh series whose coefficients satisfy conditions of bounded variation of order $(p,0), (0,p)$, and $(p,p)$ for some $p\ge 1$: (a) regular convergence; (b) uniform convergence; (c) $L^{r}$-integrability and $L^{r}$-metric convergence for $0<r<1/p$; and (d) Parseval’s formula. Extensions to those with coefficients of generalized bounded variation are also derived.