# Xhevat Z. KrasniqiUniversity of Prishtina · Faculty of Education

Xhevat Z. Krasniqi

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73

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Introduction

**Skills and Expertise**

## Publications

Publications (73)

"In this paper we have proved a theorem which show the degree of approximation of periodic functions by some generalized means of their Fourier series. In addition, our result is extended to two-dimensional setting as well."

In this paper we have generalized the classical Pell, Pell-Lucas, and Modified Pell numbers. The new numbers are named Pell, Pell-Lucas, and Modified Pell
numbers with an exponential grower factor, respectively. Moreover, we have listed their first ten terms, then we have found the families of generating functions with some of their particular grap...

In this paper we consider some deferred matrix means of Fourier series. We give the degree of seminormed approximation of functions from the spaces HP(w)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddside...

In this paper we have introduced the deferred generalized de la Vall\'ee Poussin means. Using such means of partial sums of the Fourier series of continuous and periodic functions, we have proved some theorems pertaining to upper bound of such means, to the upper bound of the their deviation from the considered functions, and to the degree of appro...

In this paper, we have proved four theorems on the degree of approximation of continuous functions by matrix means of their Fourier series which is expressed in terms of the modulus of continuity and a non-negative mediate function.

In this paper, we give a degree of approximation of a function in the space $H_{p}^{(\omega, \omega)}$ by using the second type double delayed arithmetic means of its Fourier series. Such degree of approximation is expressed via two functions of moduli of continuity type. To obtain one more general result, we used the even-type double delayed arith...

In this paper, for the first time, we introduce the deferred matrix means which contain the well-known generalized deferred Nörlund, deferred Nörlund, deferred Riesz, deferred Cesàro means introduced earlier by others, and a new class of sequences (predominantly a wider class than the class of Head Bounded Variation Sequences). In addition, using t...

In this paper, using new conditions on entries of considered matrices, we obtain some general theorems on the rate of pointwise approximation of the Lebesgue-integrable functions by modified means of the partial sums of their Fourier series and conjugate ones. The results are given in terms of weighted modulus of continuity and have essentially wid...

In this paper, we have proved four theorems on the degree of approximation of continuous functions by matrix means of their Fourier series which is expressed in terms of the modulus of continuity and a non-negative mediate function.

Using the Mean Rest Bounded Variation Sequences or the Mean Head Bounded Variation Sequences, we have proved four theorems pertaining to the degree of approximation in sup-norm of a continuous function f by general means τλn;A(f) of partial sums of its Fourier series. The degree of approximation is expressed via an auxiliary function H(t) ≥ 0 and v...

In this paper, we have studied the degree of approximation of certain bivariate functions by double
factorable matrix means of a double Fourier series. Four theorems are proved using single rest
bounded variation sequences, single head bounded variation sequences, double rest bounded variation
sequences, and two non-negative mediate functions. Thes...

In this paper, we have proved four theorems on deferred generalized De la Vallée Poussin summability of Fourier series. Some results obtained previously by others, pertaining to generalized De la Vallée Poussin summability of Fourier series, are particular cases of ours.

In this paper we define the class GMSF(α, β, γ) of General Monotone Sequence of Functions with majorants . For a sequence of admissible functions belonging to this class we find
necessary and suﬃcient conditions under which the sine integral-series
converges in the regular sense uniformly in .

Using the repeated de la Vallée Poussin sums, we have proved four theorems which show the upper bound of the repeated de la Vallée Poussin kernel, the convergence of Fourier series, the deviation between a continuous function and their repeated de la Vallée Poussin sums of partial sums of their Fourier series, and the degree of approximation of fun...

In this article, we have presented the necessary and sufficient conditions for the power integrability with a weight of the sum of sine and cosine series whose coefficients belong to the $RBVS_{+,\omega}^{r,\delta }$ class.

We show that the new complex modified trigonometric sums, introduced by others, converge in the space L under a new class of sequences.

The class of convex sequences came across in several branches of mathematics as well as their generalizations. The present paper introduces a new classes of convex sequences, the class of two-α-convex sequences of order three. Moreover, the characterization of sequences belonging to this class is shown.

In this paper we deal with a special double sine trigonometric series formed by its blocks. This type of trigonometric series is of particular interest since its blocks always are bounded, that is, under some additional assumptions the sum-function of such series always exists. We give some conditions under which such sum-function is integrable of...

In this paper, consider the trigonometric series where (cm)ϵz is a sequence of complex numbers such that (formula presented)Then the (r - 1)-th derivative of the trigonometric series converges absolutely and uniformly. If we denote the sum function of such trigonometric series by f(x), then its (r - 1)-th derivative f(r-1)(x) is obviously a continu...

In this paper we prove that the condition $\sum_{k=\left[\frac{n}{2}\right] }^{2n}\frac{\lambda _{k}(p)}{(|n-k|+1)^{2-p}}=o(1)\, \left( =O(1) \right),$ is a necessary condition for the $L^{p}(0<p<1)-$convergence (upper boundedness) of a trigonometric series. Precisely, the results extend some results of A. S. Belov.

In this paper we introduce some numerical classes of double sequences. Such classes are used to show some sufficient conditions for L¹−convergence of double sine series. This study partially extends very recent results of Leindler, and particularly those of Zhou, from single to two-dimensional sine series.

In this paper we have studied p-th power integrability of func- tions sin xg(x) and sin xf(x) with a weight, where g(x) and f(x) denote the formal sum functions of sine and cosine trigonometric series respectively. This study may be taken as a continuation for some recent foregoings re- sults proven by L. Leindler [3] and S. Tikhonov [7] employing...

In this paper we introduce an essential class of real sequences named as (p, q; r)-convex sequences. Employing this class we generalize two different results proved previously by others.

In this paper we obtain some sufficient conditions on vertical bar N, p(n), q(n)vertical bar k summability of an orthogonal series. These conditions are expressed in terms of the coefficients of the orthogonal series. Also, several known and new results are deduced as corollaries of the main results.

In this paper, we prove two theorems on |A,δ|k, 1≤k≤2,δ≥0, summability of orthogonal series. Also, several known and new results are deduced as corollaries of the main results.

Here in this paper we have introduced a new condition which is not worse than the condition that satisfy numerical sequences of Rest Bounded Variation Mean Sequences. This condition is used to obtain some integrability conditions of the functions g(x) and f(x) (which denote formal sine and cosine trigonometric series respectively) such that these f...

We give necessary conditions in terms of the coefficients for the convergence of a double trigonometric series in the Lp-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 Hp and...

In this paper we deal with a class of sine and cosine trigonometric series. We have imposed some conditions on their coefficients so that their sums will be integrable with the exception of two points. In fact, some estimates of the integrals of the absolute values of their sums, expressed in terms of the coefficients, are obtain extending some res...

The degree of approximation of functions, that belong to generalized Lipschitz classes, by Nörlund and Riesz transforms of partial sums of a trigonometric Fourier series in the weighted Lebesgue spaces is obtained.

In this paper we have introduced a new class of numerical sequences named as Mean Rest Bounded Variation Sequence of second order. This class is used to show some integrability conditions of the functions sin xg(x) and sin xf(x) such that these functions belong to the Orlicz space, where g(x) and f(x) denote formal sine and cosine trigonometric ser...

In this paper we prove some results on the rate of pointwise approximation of functions by means of some matrix transformations related to the partial sums of a Fourier series, removing the assumptions that entries of the considered matrix belong to the classes RBV S or HBV S. In fact, with weaker assumptions, our results give better degrees than t...

In this paper the q-absolute Cesàro summability of order one is introduced in order to make an advanced study in the special topic of absolute summability of orthogonal series. Moreover, we give some sufficient conditions in terms of the coefficients of an orthogonal series under which such series is q-absolute Cesàro summable almost everywhere.

In this paper we introduce some new modified cosine sums and then using these sums we study L1-convergence of trigonometric cosine series.

In this paper we prove two theorems on absolute generalized Hausdorff summability of orthogonal series. These theorems give some sufficient conditions in terms of coefficients of an orthogonal series under which it is absolute generalized Hausdorff summable. In addition, it is verified that several known results are corollaries of the new results.

In this paper we present some results on absolute almost generalized Nrlund summability of orthogonal series. The most important corollaries of the main results also are deduced.

In the paper, we prove two theorems on {pipe}A, δ{pipe}k summability, 1 ≤ k ≤, of orthogonal series. Several known and new results are also deduced as corollaries of the main results.

In this paper we deal with cosine and sine trigonometric series with generalized semi-convex coefficients. Integrability conditions for them are obtained.

We establish some theorems on the degree of approximation of continuous functions by matrix means related to partial sums of a Fourier series, employing some known and new wider classes of null-sequences than those of Rest Bounded Variation Sequences or of Head Rest Bounded Variation Sequences. These new results give significantly better degrees th...

In this paper, we prove two theorems on |A| k, 1 ≤ k ≤ 2, summability of orthogonal series. The first one gives a sufficient condition under which an orthogonal series is absolutely summable almost everywhere, and the second one, is a general theorem, which also gives a sufficient condition so that an orthogonal series is absolutely summable almost...

In this paper we obtain some sufficient conditions on |(N,p,q)(E,1)| k , (1≤k≤2), summability of an orthogonal series. These conditions are expressed in terms of the coefficients of the orthogonal series. Several important results are also deduced as corollaries.

In this paper we obtain estimates of the sum of double sine series near the origin, with multiple-monotone coefficients tending to zero. These estimates extend some results of Telyakovski [11] and Popov [7] from single to multidimensional case.

A condition of integrability of cosine trigonometric series with coefficients of bounded variation of order p is obtained. The results generalize some previous results of S. A. Telyakovskij [in: E. F. Mishchenko (ed.) et al., The theory of functions and differential equations. Collected papers. In honor of the ninetieth birthday of Academician Serg...

Some representations for the second derivatives of the sums of the cosine or sine trigonometric series are found in terms of the second differences of their coefficients. If for the cosine series we denote its sum by f(x), then it is proved that under certain conditions the function f(x)-(a 1 -2a 2 )x is concave or convex on (0,π], which demonstrat...

We extend some results of S. A. Telyakovskij [Proc. Steklov Inst. Math. 219, 372–381 (1997; Zbl 0926.42004)] from single to double trigonometric Fourier series. We study the absolute convergence of a special double trigonometric Fourier series of any function of bounded variation in the sense of Hardy and Krause. Besides that, a reinforced version...

In this paper, for the sum of sine or cosine series with quasi-convex coefficients of higher order, the representation of their first deriva-tives are found in terms of the r-th differences of coefficients of the series obtained by formal differentiation. Also some estimates in terms of coef-ficients of the series are obtained for the integrals of...

In this paper are considered the modified cosine sums introduced by Rees and Stanojević with coefficients from the class K. In addition, it is proved that the condition $lim_{no infty}|a_{n+1}|log n= 0$ is a necessary and sufficient condition for the $L^{1}$-convergence of the cosine series. Also, an open problem about $L^{1}$-convergence for the $...

We study L1-convergence of r − th derivative of modified cosine sums introduced in [2]. Exactly it is proved the L1-convergence of r − th derivative of modified cosine sums with r-quasi convex coefficients.

In this paper we present some results on |N, p, q|k, (1 ≤ k ≤ 2) summability of orthogonal series.

In this paper we prove that the conditon Sigma(2n)(k=[n/2]) k(r)lambda(k)/vertical bar n - k vertical bar + 1 = o(1) (= O(1)), for r = 0, 1, 2, ..., is necessary for the convergence of the r - th derivative of the Fourier series in the L(1)-metric. This condition is sufficient under some additional assumptions for Fourier coefficients. In fact, in...

We introduce new modified cosine and sine sums and study their L1-convergence to the sine and cosine trigonometric se- ries respectively and deduce a result of Bala and Ram (1) as a corollary.

A criterion for L 1 - convergence of a certain cosine sums with quasi semi-convex coefficients is obtained. Also a necessary and sufficient condition for L 1 -convergence of the cosine series is deduced as a corollary.

We obtained the degree of approximation of functions belonging to Lipschitz classes, with double Fourier series, using Euler means.

Let (a k ) be a numerical sequence which tends to zero and is semiconvex. Our aim is to estimate the sine series g(x):=∑ k=1 ∞ a k sinkx for x→0, expressed in terms of the coefficients a k .

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)∼mna m,n +m n∑ l=1 n-1 la m,l +n m∑ k=1 m-1 ka k,n +1 mn∑ l=1 n-1 ∑ k=1 m-1 kla k,l , where x∈(π m+1,π m], y∈(π n+1,π n],...

We obtain estimates of the r-derivative near the origin of sine series with monotone coefficients of higher order tending to zero. These results are extensions of corresponding Telyakovkii’s results for sine series (case: k=1; r=0).

In this paper we obtain a generalization of Lipschitz's classes m(,p,r ) defined in (1). We give necessary conditions for even or odd functions with Fourier series to belong to the classes m(p,r, ). We also give sufficient conditions for even or odd functions with Fourier series to belong to the same classes.

In this paper some estimates for sine series with monotone coefficients of higher order k near the origin are given.These results generalize some results given by S.A.Telyakovskii in [2], [4].

In this paper we will give the behavior of the r derivative near origin of sine series with convex coefficients.