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 $x\in (\frac{\pi }{m+1},\frac{\pi }{m}]$, $y\in (\frac{\pi }{n+1},\frac{\pi }{n}]$, $m,n=1,2,\cdots$.

Since the trigonometric Fourier series of an integrable function does not necessarily converge to the function in the mean, several additional conditions have been devised to guarantee the convergence. For instance, sufficient conditions can be constructed by using the Fourier coefficients or the integral modulus of the corresponding function. In this paper we give a Hardy-Karamata type Tauberian condition on the Fourier coefficients and prove that it implies the convergence of the Fourier series...

Let a single sine series (*) ${\sum }_{k=1}^{\infty }{a}_{k}sinkx$ be given with nonnegative coefficients $a_k$. If $a_k$ is a “mean value bounded variation sequence” (briefly, MVBVS), then a necessary and sufficient condition for the uniform convergence of series (*) is that $k{a}_k\to 0$ as k → ∞. The class MVBVS includes all sequences monotonically decreasing to zero. These results are due to S. P. Zhou, P. Zhou and D. S. Yu. In this paper we extend them from single to double sine series (**) ${\sum }_{k=1}^{\infty }{\sum }_{l=1}^{\infty }{c}_{kl}sinkxsinly$, even with complex coefficients $c_{kl}$. We also give a uniform...

We study the class of singular measures whose Fourier partial sums converge to 0 in the metric of the weak $L^1$ space; symmetric sets of constant ratio occur in an unexpected way.