AMS Subject Classification 2010: 41A25, 41A27, 41A35, 41A36, 41A40, 42Al6, 42A85.The paper is concerned with establishing direct estimates for convolution operators on homogeneous Banach spaces of periodic functions by means of appropriately defined Kfunctional. The differential operator in the K-functional is defined by means of strong limit and described explicitly in terms of its Fourier coefficients. The description is simple and independent of the homogeneous Banach space. Saturation of such...

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...