Double Walsh series with coefficients of bounded variation of higher order

Let D-j(k)(x) denote the Cesaro 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 equal to or greater than 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.