Fejer Operator Research Articles

Best Approximation of Unbounded Functions by Modulus of Smoothness

In this paper, we study the approximation of unbounded functions in a weighted space by modulus of smoothness using various linear operators. We establish direct theorems for such approximations and analyze the properties of the modulus of smoothness within the same space. Specifically, we investigate the behavior of the modulus of smoothness under different types of linear operators, including the Bernstein-Durrmeyer operator, the Fejer operator, and the Jackson operator. We also provide a detailed analysis of the convergence rate of these operators. Furthermore, we discuss the relationship between the modulus of smoothness and the Lipschitz constant of a function. Our findings have important implications for the field of approximation theory and may help to inform future research in this area.

Approximation of functions in the Lp.w (X)-space and application

The purpose of this research is to study the approximation of functions in the space Lp.w which is an expansion of the Lebesgue space Lp using the Fejer operator in terms of the average modulus of the second-order with the application of two of the theorems.

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.

Iterative Fejér Processes in Ill-Posed Problems

A brief survey is given concerning iterative processes of Fejer type for basic statements of ill-posed problems, including constrained quadratic and convex minimization problems, variational inequalities, and linear and nonlinear operator equations in Hilbert spaces. By applying the method of successive approximations and its modification using correction factors, all these statements reduce to the problem of finding fixed points of nonexpansive Fejer operators. Material is also presented related to a two-stage method of constructing a regularizing algorithm for nonlinear ill-posed problems with monotone operators. An economic way is described by which the algorithm takes into account additional a priori information on the solution using Fejer maps.

Average error bounds of trigonometric approximation on periodic Wiener spaces

In this paper, we study the approximation of identity operator and the convolution integral operator Bm by Fourier partial sum operators, Fejer operators, Vallee-Poussin operators, Cesaro operators and Abel mean operators, respectively, on the periodic Wiener space (C1(ℝ),Wℴ) and obtain the average error estimations.

Learning Rates for Regularized Classifiers Using Trigonometric Polynomial Kernels

Regularized classifiers are known to be a kind of kernel-based classification methods generated from Tikhonov regularization schemes, and the trigonometric polynomial kernels are ones of the most important kernels and play key roles in signal processing. The main target of this paper is to provide convergence rates of classification algorithms generated by regularization schemes with trigonometric polynomial kernels. As a special case, an error analysis for the support vector machines (SVMs) soft margin classifier is presented. The norms of Fejer operator in reproducing kernel Hilbert space and properties of approximation of the operator in L 1 space with periodic function play key roles in the analysis of regularization error. Some new bounds on the learning rate of regularization algorithms based on the measure of covering number for normalized loss functions are established. Together with the analysis of sample error, the explicit learning rates for SVM are also derived.

Hilbert, Dirichlet and Fejér families of operators arising from C 0-groups, cosine functions and holomorphic semigroups

The main aim of this paper is to extend definitions of Hilbert transform, Dirichlet and Fejer operators (defined by convolution with suitable kernels in Lebesgue spaces) in arbitrary Banach spaces. We present a self-contained theory which includes different approaches of other authors whose starting points were usually C 0-groups or cosine functions. We present relations with holomorphic semigroups. We characterize the geometric property of UMD spaces in terms of the Dirichlet and Fejer operators. To end the paper, we give examples to illustrate our results.

The use of additional diminishing disturbances in Fejer models of iterative algorithms

The behavior of Fejer processes with diminishing disturbances generated by a small shift in the argument of the Fejer operator is studied. It is shown that, if the operator is locally strongly Fejer, a diminishing disturbance does not prevent convergence to an attracting set. At the same time, such a disturbance can be used to furnish the process with additional properties that ensure convergence to certain subsets of the attracting set. In particular, based on this scheme, new parallel decomposition methods for optimization problems can be suggested that do not require that the constraints possess a specific structure.

-summability of Fourier series

A general summability method of orthogonal series is given with the help of an integrable function Θ. Under some conditions on Θ we show that if the maximal Fejer operator is bounded from a Banach space X to Y, then the maximal Θ-operator is also bounded. As special cases the trigonometric Fourier, Walsh, Walsh--Kaczmarz, Vilenkin and Ciesielski--Fourier series and the Fourier transforms are considered. It is proved that the maximal operator of the Θ-means of these Fourier series is bounded from H p to L p (1/2<p≤; ∞) and is of weak type (1,1). In the endpoint case p=1/2 a weak type inequality is derived. As a consequence we obtain that the Θ-means of a function f∈L 1 converge a.e. to f. Some special cases of the Θ-summation are considered, such as the Weierstrass, Picar, Bessel, Riesz, de la Vallee-Poussin, Rogosinski and Riemann summations. Similar results are verified for several-dimensional Fourier series and Hardy spaces.

The maximal conjugate Fej�r operator is bounded from L 1 to weak- L 1

We consider the Fejer (or first arithmetic) means of the conjugate series to the Fourier series of a periodic function f integrable in Lebesgue's sense on the torus \({\Bbb T }:= [-\pi , \pi )\). A classical theorem of A. Zygmund says that the maximal conjugate Fejer operator \(\widetilde \sigma _*(f)\) is bounded from \(L^1({\Bbb T })\) to \(L^p({\Bbb T })\) for any \(0\le p \le 1\). We sharpen this result by proving that \(\widetilde \sigma _*(f)\) is bounded from \(L^1({\Bbb T })\) to weak-\(L^1({\Bbb T })\). We prove an analogous result also for the Fejer means (or Riesz means of first order) of the conjugate integral to the Fourier integral of a function f integrable in Lebesgue's sense on the whole real line \({\Bbb R}:= (-\infty , \infty )\).