A two-dimensional extension of Zygmund's theorem on the smoothness of the sum of trigonometric series

Given a double sequence {c(m,n) : (m, n) is an element of Z(2)} of complex numbers, we consider the double trigonometric series (*) Sigma(m is an element of Z) Sigma(n is an element of Z) c(m,n)e(i(mx+ny)), which converges absolutely and uniformly, thus its sum f (x, y) is continuous. We give sufficient conditions in terms of certain means of {c(m,n)} to guarantee that f (x, y) belongs to one of the Zygmund classes Zyg(alpha, beta) and zyg(alpha, beta) for some 0 < alpha, beta <= 2. The present theorems extend those in [3] from single to double trigonometric series, the latter ones in turn were the generalizations of the corresponding theorem of Zygmund in [5]. Our method of proof is essentially different from that of Zygmund. We establish four lemmas, which reveal interrelations between the order of magnitude of certain initial means and that of certain tail means of the double sequence {c(m, n)}.