Fejer Means Research Articles

Approximation of classes of Poisson integrals by Fejer means

The paper is devoted to the investigation of problem of approximation of continuous periodic functions by trigonometric polynomials, which are generated by linear methods of summation of Fourier series.
 The simplest example of a linear approximation of periodic functions is the approximation of functions by partial sums of their Fourier series. However, the sequences of partial Fourier sums are not uniformly convergent over the class of continuous periodic functions. Therefore, many studies devoted to the research of the approximative properties of approximation methods, which are generated by transformations of the partial sums of Fourier series and allow us to construct sequences of trigonometrical polynomials that would be uniformly convergent for the whole class of continuous functions. Particularly, Fejer sums have been widely studied recently. One of the important problems in this area is the study of asymptotic behavior of the sharp upper bounds over a given class of functions of deviations of the trigonometric polynomials.
 In the paper, we study upper asymptotic estimates for deviations between a function and the Fejer means for the Fourier series of the function. The asymptotic behavior is considered for the functions represented by the Poisson integrals of periodic functions of a real variable. The mentioned classes consist of analytic functions of a real variable. These functions can be regularly extended into the corresponding strip of the complex plane.An asymptotic equality for the upper bounds of Fejer means deviations on classes of Poisson integrals was obtained.

Approximation of classes of Poisson integrals by Fejer means

The work is devoted to the investigation of problem of approximation of continuous periodic functions by trigonometric polynomials, which are generated by linear methods of summation of Fourier series. The simplest example of a linear approximation of periodic functions is the approximation of functions by partial sums of their Fourier series. However, the sequences of partial Fourier sums are not uniformly convergent over the class of continuous periodic functions. Therefore, a many studies is devoted to the research of the approximative properties of approximation methods, which are generated by transformations of the partial sums of Fourier series and allow us to construct sequences of trigonometrical polynomials that would be uniformly convergent for the whole class of continuous functions. Particularly, Fejer means have been widely studied in the last time. One of the important problems in this field is the study of asymptotic behavior of the upper bounds over a fixed classes of functions of deviations of the trigonometric polynomials. The aim of the work systematizes known results related to the approximation of classes of Poisson integrals of continuous functions by arithmetic means of Fourier sums, and presents new facts obtained for particular cases. The asymptotic behavior of the upper bounds on classes of Poisson integrals of periodic functions of the real variable of deviations of linear means of Fourier series, which are defined by applying the Fejer summation method is studied. The mentioned classes consist of analytic functions of a real variable, which are narrowing of bounded harmonic in unit disc functions of complex variable. In the work, asymptotic equalities for the upper bounds of deviations of Fejer means on classes of Poisson integrals were obtained.

Vilenkin–Lebesgue Points and Almost Everywhere Convergence for Some Classical Summability Methods

The concept of Vilenkin–Lebesgue points was introduced in [12], where the almost everywhere convergence of Fejer means of Vilenkin–Fourier series was proved. In this paper, we present a different (and simpler) approach to prove a similar result, which can be used to prove that the corresponding result holds also in a more general context, namely for regular Norlund and T-means.

Martingale Musielak–Orlicz–Lorentz Hardy Spaces with Applications to Dyadic Fourier Analysis

Let $$(\Omega ,{\mathcal {F}},{\mathbb {P}})$$ be a probability space, $$\varphi :\ \Omega \times [0,\infty )\rightarrow [0,\infty )$$ a Musielak–Orlicz function, and $$q\in (0,\infty ]$$ . In this article, the authors introduce five martingale Musielak–Orlicz–Lorentz Hardy spaces and prove that these new spaces have some important features such as atomic characterizations, the boundedness of $$\sigma $$ -sublinear operators, and martingale inequalities. This new scale of martingale Hardy spaces requires the introduction of the Musielak–Orlicz–Lorentz space $$L^{\varphi ,q}(\Omega )$$ . In particular, the authors show that this Lorentz type space has some fundamental properties including the completeness, the convergence, real interpolations, and the Fefferman–Stein vector-valued inequality for the Doob maximal operator. As applications, the authors prove that the maximal Fejer operator is bounded from the martingale Musielak–Orlicz–Lorentz Hardy space $$H_{\varphi ,q}[0,1)$$ to $$L^{\varphi ,q}[0,1)$$ , which further implies some convergence results of the Fejer means. Moreover, all the above results are new even for Musielak–Orlicz functions with particular structure such as weight, weight Orlicz, and double-phase growth. The main approach used in this article can be viewed as a combination of the stopping time argument in probability theory and the real-variable technique of function spaces in harmonic analysis.

Conjugate Transforms on Dyadic Group

In this paper we study the properties of the Lebesgue constant of the conjugate transforms. For conjugate Fejer means we will find necessary and sufficient condition on $$t$$ for which the estimation $$E\left|\widetilde{\sigma}_{n}^{\left(t\right)}f\right|\leq cE\left|f\right|$$ holds. We also prove that for dyadic irrational $$t$$ , $$L\log L$$ is the maximal Orlicz space for which the estimation $$E\left|\widetilde{\sigma}_{n}^{\left(t\right)}f\right|\leq c_{1}+c_{2}E\left(\left|f\right|\log^{+}\left|f\right|\right)$$ is valid.

On the boundedness of subsequences of Vilenkin-Fejér means on the martingale Hardy spaces

In this paper we characterize subsequences of Fejer means with respect to Vilenkin systems, which are bounded from the Hardy space H-p to the Lebesgue space L-p, for all 0 < p < 1/2. The resu ...

Fejer means of rational Fourier – Chebyshev series and approximation of function |x|s

Approximation properties of Fejer means of Fourier series by Chebyshev – Markov system of algebraic fractions and approximation by Fejer means of function |x|s, 0 &lt; s &lt; 2, on the interval [−1,1], are studied. One orthogonal system of Chebyshev – Markov algebraic fractions is considers, and Fejer means of the corresponding rational Fourier – Chebyshev series is introduce. The order of approximations of the sequence of Fejer means of continuous functions on a segment in terms of the continuity module and sufficient conditions on the parameter providing uniform convergence are established. A estimates of the pointwise and uniform approximation of the function |x|s, 0 &lt; s &lt; 2, on the interval [−1,1], the asymptotic expressions under n→∞ of majorant of uniform approximations, and the optimal value of the parameter, which provides the highest rate of approximation of the studied functions are sums of rational use of Fourier – Chebyshev are found.

Approximation of Bounded Holomorphic and Harmonic Functions by Fejér Means

We compute the exact values of the least upper bounds on the classes of bounded holomorphic and harmonic functions in a unit disk for the remainders in a Voronovskaya-type formula in the case of approximation by Fejer means. We also present some consequences that are of independent interest.

On derivative of trigonometric polynomials and characterizations of modulus of smoothness in weighted Lebesgue space with variable exponent

In this paper we investigate some properties of approximation polynomials in particular de la Vallee-Poussin means, Fejer means and partial sums of Fourier series in weighted Lebesgue spaces with variable exponent. In addition to these we prove a simultaneous type theorem and some theorems on the equivalence of modulus of smoothness and the K-functional in weighted Lebesgue space with variable exponent.

Uniform Approximation of the Curvature of Smooth Plane Curves with the Use of Partial Fourier Sums

An error bound for the approximation of the curvature of graphs of periodic functions from the class Wr for r ≥ 3 in the uniform metric is obtained with the use of the simplest approximation technique for smooth periodic functions, which is approximation by partial sums of their trigonometric Fourier series. From the mathematical point of view, the interest in this problem is connected with the specific nonlinearity of the graph curvature operator on the class of smooth functions Wr on a period or a closed interval for r ≥ 2. There are several papers on curvature approximation for plane curves in the mean-square and Chebyshev norms. In previous works, the approximation was performed by partial sums of trigonometric series (in the L2 norm), interpolation splines with uniform knots, Fejer means of partial sums of trigonometric series, and orthogonal interpolating wavelets based on Meyer wavelets (in the C∞ norm). The technique of this paper, based on the lemma, can possibly be generalized to the Lp metric and other approximation methods.

Summation of Fourier Series on the Infinite-Dimensional Torus

The conditions for convergence of square and rectangular Fejer means of functions on the infinite dimensional torus were obtained, also a generalization of the results for the case of abstract measure spaces was formulated.

Local stress in periodic composites via the Riesz summability method

Abstract Different summation methods of multidimensional trigonometric Fourier series are adopted for approximating the local strain and stress in elastic periodic composites. In fact, approximating the exact solution of a problem with the original partial sums of Fourier series can produce unwanted effects, such as the Gibbs phenomenon at the jump discontinuities that may occur at the interface between the constituents of the composites. Nevertheless, the Fourier coefficients of the original partial sums contain enough information to provide better estimates via the summability methods. In the means provided by the summability methods, the Fourier coefficients are suitably weighted by reducing windows, involving a regularization of the numerical solution. First, a complete solution method is used in order to determine some Fourier coefficients of the local strain in periodic composites. Next, these Fourier coefficients are used to construct suitable summations and means exhibiting better convergence properties than those of the original partial sums of Fourier series. The proposed method, valid for a great variety of the geometry of the constituents embedded in the matrix, is here applied to unidirectional composites with cylindrical fibers. The behavior of the iterated Fejer partial sums, Fejer and Riesz means is investigated in order to improve the convergence, observing that the Riesz means have better convergence properties than those of the original partial sums of Fourier series.

On the convergence of Fejér means of Walsh-Fourier series in the space H p

The main aim of this paper is to find necessary and sufficient conditions for a modulus of continuity of a martingale F ∈ Hp, for which the Fejer means of Walsh-Fourier series converge in Hp-norm, when 0 < p ≤ 1/2.

On the restricted summability of two-dimensional Walsh--Fejer means

On the restricted summability of two-dimensional Walsh--Fejer means

Strong convergence theorem for Vilenkin--Fejer means

As main result we prove strong convergence theorems of Vilenkin–Fejer means when 0 < p ≤ 1/2.

Approximation of holomorphic functions of Zygmund class by Fejér means

We obtain an asymptotic equality for the upper bounds of deviations of Fejer means in the Zygmund class of functions holomorphic in the unit disk.

Almost everywhere convergence of sequences of two-dimensional Walsh–Fejér means of integrable functions

The aim of this paper is to prove the a.e. convergence of sequences of the Fejer means of the Walsh–Fourier series of bivariate integrable functions. That is, let \(a = (a_{1}, a_{2})\:\mathbb{N} \to \mathbb{N}^{2}\) such that a j (n+1)≧δsup k≦n a j (n) (j=1,2, n∈ℕ) for some δ>0 and a 1(+∞)=a 2(+∞)=+∞. Then for each integrable function f∈L 1(I 2) we have the a.e. relation \(\lim_{n\to\infty}\sigma_{a_{1}(n), a_{2}(n)}f = f\). It will be a straightforward and easy consequence of this result the cone restricted a.e. convergence of the two-dimensional Walsh–Fejer means of integrable functions which was proved earlier by the author and Weisz [3,8].

Pointwise summability of Vilenkin--Fourier series

In this paper we give a characterization of points in which Fejer means of Vilenkin-Fourier series converge.

Pointwise convergence of cone-like restricted two-dimensional Fejér means of Walsh-Fourier series

For the two-dimensional Walsh system, Gat and Weisz proved the a.e. convergence of Fejer means σnf of integrable functions, where the set of indices is inside a positive cone around the identical function, that is, β−1 ≤ n1/n2 ≤ β is provided with some fixed parameter β ≥ 1. In this paper we generalize the result of Gat and Weisz. We not only generalize this theorem, but give a necessary and sufficient condition for cone-like sets in order to preserve this convergence property.

Restricted maximal operators of Fejér means of double Walsh-Fourier series

The main aim of this paper is to prove that there exists a martingale f ∈ H12/▭ such that the restricted maximal operators of Fejer means of twodimensional Walsh-Fourier series and conjugate Walsh-Fourier series does not belong to the space weak-L1/2.