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In this paper integrability condition for double cosine trigonometric series with coefficients of bounded variation of second order has been obtained. This condition is given explicitly in terms of the coefficients. The result extends a previous result of Telyakovskiı̆ from one dimensional case to two dimensional case.

In this paper we generalize some results on the degree of approximation of continuous functions by matrix means related to partial sums of a Fourier series, obtained previously by some other authors (please consult references cited in this paper).

In this paper we introduce some new modified cosine sums and then using these sums we study $L^1$-convergence of trigonometric cosine series.

In the paper, we prove two theorems on ${|A,\delta |}_k$ summability, $1\le k\le 2$, of orthogonal series. Several known and new results are also deduced as corollaries of the main results.

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 $a_{k,l}$ satisfy certain conditions) the following order equality is proved $$g(x,y)\sim mn{a}_{m,n}+\frac{m}{n}\sum _{l=1}^{n-1}l{a}_{m,l}+\frac{n}{m}\sum _{k=1}^{m-1}k{a}_{k,n}+\frac{1}{mn}\sum _{l=1}^{n-1}\sum _{k=1}^{m-1}kl{a}_{k,l},$$ where $x\in (\frac{\pi }{m+1},\frac{\pi }{m}]$, $y\in (\frac{\pi }{n+1},\frac{\pi }{n}]$, $m,n=1,2,\cdots$.

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 $H^p$ and the Bernstein-Zygmund inequalities for the derivatives of trigonometric polynomials and their conjugates in the $L^p$-metric, where $0<p<1$.

MSC 2010: 42A32; 42A20 In this paper we have defined a new class of double numerical sequences. If the coefficients of a double cosine or sine trigonometric series belong to the such classes, then it is verified that they are Fourier series or equivalently their sums are integrable functions. In addition, we obtain an estimate for the mixed modulus of smoothness of a double sine Fourier series whose coefficients belong to the new class of sequences mention above.