A new class of numerical sequences and its applications to uniform convergence of sine series

In the present paper we introduce a new class of sequences called GM(b,r), which is the generalization of a class considered by Tikhonov. Moreover, we obtained in this note sufficient and necessary conditions for uniform convergence of sine series with (b,r)-general monotone coefficients

Approximation of conjugate functions by general linear operators of their Fourier series at the Lebesgue points

The pointwise estimates of the deviations \widetilde{T}_{n,A,B}^{\text{}%}f\left(\cdot \right) -\widetilde{f}(\cdot) and \widetilde{T}_{n,A,B}^{% \text{}}f\left(\cdot \right) -\widetilde{f}(\cdot,\varepsilon) in terms of moduli of continuity $\widetilde{\bar{w}}_{\cdot}f$ and $\widetilde{w}$ are proved. Analogical results on norm approximation with remarks and corollary are also given. These results generalized a theorem of Mittal

Estimates of convolution operators of functions from L2πp(x)L_{2\pi}^{p(x)}

We generalize and slight improve the result of I. I. Sharapudinov [Mat. Zametki, 1996, Volume 59, Issue 2, 291--302]. Some applications to the de la Vall\'{e}e Poussin operator will also be given

On the degree of strong approximation of almost periodic functions in the Stepanov sense

Considering the class of almost periodic functions in the Stepanov sense we extend and generalize the results of the first author [4]. as well as the results of L. Leindler [3] and P. Chandra [1,2]

Degree of Convergence of an Integral Operator

In this paper we define an integral operator on Lp and obtain its degree of convergence in the appropriate norm. By specializing the kernel of the integral operator we obtain many known results as corollaries. We have also applied our results to obtain results on singular integral operators

On some Leindler's theorem on application of the class NMCS

We show the results in the class GM(5b) corresponding to the theorem of L. Leindler [A note on strong approximation of Fourier series, Analysis Mathematica, 29(2003), 195--199] on strong approximation by matrix means of Fourier series constructed by the sequences from the class NMCS

Strong approximation of almost periodic functions

We consider summability methods generated by the class GM(2b). We generalize some related results of P. Pych-Taberska [Studia Math. XCVI (1990), 91-103] on strong approximation of almost periodic functions by their Fourier series and S. M. Mazhar and V. Totik [J. Approx. Theory, 60(1990), 174-182] on approximation of periodic functions by matrix means of their Fourier series

On the uniform convergence of double sine series

The fundamental theorem in the theory of the uniform convergence of sine series is due to Chaundy and Jolliffe from 1916 (see [1]). Several authors gave conditions for this problem supposing that coefficients are monotone, non-negative or more recently, general monotone (see [8], [6] and [3], for example). There are also results for the regular convergence of double sine series to by uniform in case the coefficients are monotone or general monotone double sequences. In this article we give new sufficient conditions for the uniformity of the regular convergence of double sine series, which are necessary as well in case the coefficients are non-negative. We shall generalize those results defining a new class of double sequences for the coefficients.Comment: 20 page

Pointwise approximation of functions by matrix operators of their Fourier series with rr- differences of the entries

We extend the results of Xh. Z. Krasniqi [Acta Comment. Univ. Tartu. Math. 17 (2013), 89-101] and the authors [Acta Comment. Univ. Tartu. Math. 13 (2009), 11-24]. to the case where in the measures of estimations there are used $r$-differences of the entries

On trigonometric approximation of functions in the Lp norm

In this paper we obtain degree of approximation of functions in Lp by operators associated with their Fourier series using integral modulus of continuity. These results generalize many know results and are proved under less stringent conditions on the infinite matrix