ACTA issues

Lebesgue integrability theorems for double trigonometric series

Kitti F. Papp

Acta Sci. Math. (Szeged) 61:1-4(1995), 331-344

Abstract. We study double cosine, double sine, and cosine-sine series whose coefficients $\{a_{jk}\} $ are such that $\sum_{j=0}^\infty\sum _{k=0}^\infty |a_{jk}|< \infty $. Then each of these series converges uniformly to some sum denoted by $f(x,y)$, $g(x,y)$, and $h(x,y)$, respectively. We give sufficient conditions for the $L^1$-integrability of the quotients $[f(x,y)- f(x,0)-f(0,y)+f(0,0)]/xy$, $g(x,y)/xy$, and $[h(x,y)-h(0,y)]/xy$. Our theorems extend those of F. Móricz from the one-dimensional to two-dimensional series.

AMS Subject Classification (1991): 42B99, 42A16

Received May 2, 1995. (Registered under 5689/2009.)