Some new properties of the stationary sets (defined by G. Pisier in [12]) are studied. Some arithmetical conditions are given, leading to the non-stationarity of the prime numbers. It is shown that any stationary set is a set of continuity. Some examples of "large" stationary sets are given, which are not sets of uniform convergence.

Let m: ℝ → ℝ be a function of bounded variation. We prove the $L^p\left(ℝ\right)$-boundedness, 1 < p < ∞, of the one-dimensional integral operator defined by $T_{m}f\left(x\right)=p.v.\int k(x-y)m(x+y)f\left(y\right)dy$ where $k\left(x\right)={\sum }_{j\in \mathbb{Z}}{2}^j{\phi }_j\left(2^{j}x\right)$ for a family of functions ${{\phi }_j}_{j\in \mathbb{Z}}$ satisfying conditions (1.1)-(1.3) given below.

We study different conditions on the set of roots of the Fourier transform of a measure on the Euclidean space, which yield that the measure is absolutely continuous with respect to the Lebesgue measure. We construct a monotone sequence in the real line with this property. We construct a closed subset of ${ℝ}^d$ which contains a lot of lines of some fixed direction, with the property that every measure with spectrum contained in this set is absolutely continuous. We also give examples of sets with such property...

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...

We generalize and improve in some cases the results of Mahapatra and Chandra [7]. As a measure of Hölder norm approximation, generalized modulus-type functions are used.

Under some assumptions on the matrix of a summability method, whose rows are sequences of bounded variation, we obtain a generalization and an improvement of some results of Xie-Hua Sun and L. Leindler.

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