We give necessary conditions in terms of the coefficients for the convergence of a double trigonometric series in the L p -metric, where 0 < p < 1. The results and their proofs have been motivated by the recent papers of A. S. Belov (2008) and F. Móricz (2010). Our basic tools in the proofs are the Hardy-Littlewood inequality for functions in Hp and the Bernstein-Zygmund inequalities for the derivatives of trigonometric polynomials and their conjugates in the L p -metric, where 0 < p < 1.
In this paper we obtain estimates of the sum of double sine series near the origin, with monotone coefficients tending to zero. In particular (if the coefficients ak,l satisfy certain conditions) the following order equality is proved g(x,y) ∼ mnam,n + m⁄ n ∑ n−1 l=1 lam,l + n⁄ m ∑mX−1 k=1 kak,n + 1 m ⁄n ∑ n−1 l=1 ∑ m−1 k=1 klak,l, where x ∈ ( π⁄ m+1 , π ⁄ m ], y ∈ ( π ⁄ n+1 , π ⁄ n ], m, n = 1, 2, . . ..
In the paper, we prove two theorems on |A, δ|k summability, 1 ≤ k ≤ 2, of orthogonal series. Several known and new results are also deduced as corollaries of the main results.