Integrability of double trigonometric series with special coefficients (Q1813305)

Double cosine- and sine-series are considered and also series with respect to mixed harmonics, the coefficients of which tend to zero and constitute a sequence of bounded variation. Along with rectangular partial sums of such series modified sums are investigated, and necessary and sufficient conditions for their regular convergence in the metric of \(L^ 1=L^ 1\bigl( [0,\pi] \times [0,\pi] \bigr)\) are established. As corollaries, coefficient conditions for the integrability of the sum of a double trigonometric series are established; under the same conditions it is the Fourier series of its sum, and the rectangular partial sums converge regularly in the metric of \(L^ 1\). We note that for one- dimensional cosine-series similar modified sums were introduced and considered by \textit{J. W. Garrett} and \textit{Č. V. Stanojević} [Proc. Am. Math. Soc. 60, 68-71 (1976; Zbl 0339.42007)], and analogous results for them were established by these authors; one-dimensional sine-series were considered by the author in Stud. Math. 92, No. 2, 187-200 (1989; Zbl 0571.42006).