INTEGRABILITY, MEAN CONVERGENCE, AND PARSEVAL'S FORMULA FOR DOUBLE TRIGONOMETRIC SERIES

Abstract

Consider the double trigonometric series whose coefficients satisfy conditions of bounded variation of order $(p, 0)$, $(0, p)$, and $(p, p)$ with the weight $(\overline{|j|}\, \overline{|k|})^{p-1}$ for some $p>1$. The following properties concerning the rectangular partial sums of this series are obtained: (a) regular convergence; (b) uniform convergence; (c) weighted $L^r$-integrability and weighted $L^r$-convergence; and (d) Parseval's formula. Our results generalize Bary [1, p. 656], Boas [2, 3], Chen [6, 7], Kolmogorov [9], Marzug [10], M\'oricz [11, 12, 13, 14], Ul'janov [15], Young [16], and Zygmund [17, p. 4].