Vallee Poussin Research Articles

De la Vallée Poussin filtered polynomial approximation on the half–line

On the half line, we introduce a new sequence of near-best uniform approximation polynomials, easily computable by the values of the approximated function at a truncated number of Laguerre zeros. Such approximation polynomials come from a discretization of filtered Fourier–Laguerre partial sums, which are filtered using a de la Vallée Poussin (VP) filter. They have the peculiarity of depending on two parameters: a truncation parameter that determines how many of the n Laguerre zeros are considered, and a localization parameter, which determines the range of action of the VP filter we will apply. As n→∞, under simple assumptions on such parameters and the Laguerre exponents of the involved weights, we prove that the new VP filtered approximation polynomials have uniformly bounded Lebesgue constants and uniformly convergence at a near–best approximation rate, for any locally continuous function on the semiaxis.The numerical experiments have validated the theoretical results. In particular, they show a better performance of the proposed VP filtered approximation versus the truncated Lagrange interpolation at the same nodes, especially for functions a.e. very smooth with isolated singularities. In such cases, we see a more localized approximation and a satisfactory reduction of the Gibbs phenomenon.

Об ограниченности в средних Валле-Пуссена для дискретных сумм Фурье по многочленам, ортогональным на неравномерных сетках

Об ограниченности в средних Валле-Пуссена для дискретных сумм Фурье по многочленам, ортогональным на неравномерных сетках

Applications of the deferred generalized de la Vallee Poussin means in approximation of continuous functions

"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."

Negativity of Green’s Functions to Focal and Two-Point Boundary Value Problems for Equations of Second Order with Delay and Impulses in Their Derivatives

We consider the second-order impulsive differential equation with impulses in derivative and without the damping term. Sufficient conditions that a nontrivial solution of the homogeneous equation having a zero of its derivative does not have a zero itself are obtained. On the basis of the obtained results on differential inequalities, which can be considered as analogues of the Vallee–Poussin theorems, new sufficient conditions on the negativity of Green’s functions are obtained.

Grüss and Grüss-Voronovskaya-type estimates for complex convolution polynomial operators

A classical well-known result in approximation theory is the Grüss inequality for positive linear functionals, which gives an upper bound for the Chebyshev-type functional. Starting also from a problem posed by Gavrea, this inequality was also investigated in terms of the least concave majorants of the moduli of continuity and for positive linear operators by Acu, Gonska, Rasa and Rusu, where the cases of classical Hermite-Fejer and Fejer-Korovkin convolution operators were considered. Refined versions of the Grüss-type inequality in the spirit of Voronovskaya's theorem were obtained by Gal and Gonska for Bernstein and Paltanea operators of real variable and for complex Bernstein, genuine Bernstein-Durrmeyer and Bernstein-Faber operators attached to analytic functions of complex variable. After the appearance of these papers, several papers by other authors have developed these directions of research. The goal of this paper is to continue the above mentioned directions of research, obtaining Grüss and Grüss-Voronovskaya exact estimates (with respect to the degree of polynomials) for the de la Vallee-Poussin complex polynomials in Section 2, for Zygmund-Riesz complex polynomials in Section 3 and for Jackson complex polynomials in Section 4.

О приближении функций класса Зигмунда бигармоническими интегралами Пуассона

Работа посвящена изучению поведения верхнего предела отклонения функций класса Зигмунда от их бигармонических интегралов Пуассона. Исследования в данном направлении проводились и проводятся систематически как отечественными, так и зарубежными учеными. Большинство полученных результатов относится к оценке отклонений функций того или иного класса от операторов, построенных с помощью треугольных методов суммирования рядов Фурье (Фейера, Валле Пуссена, Рисса, Рогозинского, Стеклова, Фавара и др.). Что касается результатов относительно линейных методов суммирования рядов Фурье, заданных с помощью множества функций натурального аргумента (Абеля-Пуассона, Гаусса-Вейерштрасса, бигармонического и трехгармонического интегралов Пуассона), то здесь успехи менее заметны. Возможно, это связано с тем, что упомянутые выше линейные методы суммирования рядов Фурье являются решениями соответствующих интегрально-дифференциальных уравнений эллиптического типа и поэтому требуют более трудоемких вычислений с целью получения для них определенных оценок, пригодных для их непосредственное использование в прикладных целях. Исследования, проведенные в данной работе, относятся к изучению апроксимативных характеристик линейных положительных операторов типа Пуассона на классах функций Зигмунда. Согласно хорошо известным результатам П.П. Коровкина именно эти положительные линейные операторы осуществляют наилучшее асимптотическое приближение функций класса Зигмунда. Таким образом, полученная в данной работе оценка отклонения функций класса Зигмунда от их бигармонических интегралов Пуассона (наименее исследованных и наиболее востребованных среди всех линейных положительных операторов) актуальна с точки зрения прикладной математики.

Iterates of convolution-type operators

This paper is devoted to the study of the operators L having the property that there exists $$c\in [0,1)$$ with $$|Lf|_{Lip}\le c|f|_{Lip}$$ . A class of convolution type operators acting on spaces of continuous periodic functions for which there exists the above mentioned constant c is described. We investigate the asymptotic behaviour of the iterates of such operators and give rates of convergence of the iterates toward the limit. Some examples involving the convolution operators of de la Vallee Poussin, Jackson and Fejer–Korovkin are presented. The well-known connection between de la Vallee Poussin operators and Durrmeyer operators with Chebyshev weights is used in order to obtain new estimates of the rates of convergence of the iterates of these operators.

Compactness in quasi-Banach function spaces with applications to $$L^1$$ of the semivariation of a vector measure

We characterize the relatively compact subsets of the order continuous part $$E_a$$ of a quasi-Banach function space E showing that the strong connection between compactness, uniform absolute continuity, uniform integrability, almost order boundedness and L-weak compactness that appears in the classical setting of Lebesgue spaces remains almost invariant in this new context under mild assumptions. We also present a de la Vallee–Poussin type theorem in this context that allows us to locate each compact subset of $$E_a$$ as a compact subset of a smaller quasi-Banach Orlicz space $$E^\varPhi .$$ Our results generalize the previous known results for the Banach function spaces $$L^1(m)$$ and $$L^1_w(m)$$ associated to a vector measure m and moreover they can also be applied to the quasi-Banach function space $$L^1\left( \Vert m \Vert \right) $$ associated to the semivariation of m.

Some ideal convergent multiplier sequence spaces using de la Vallee Poussin mean and Zweier operator

We introduce multiplier type ideal convergent sequence spaces, using Zweier transform and de la Vallee Poussin mean. We study some topological and algebraic properties of these spaces. Further we prove some inclusion relations related to these spaces.

APPROXIMATION OF CLASSES OF POISSON INTEGRALS BY REPEATED FEJER SUMS

The paper is devoted to the approximation by arithmetic means of Fourier sums of classes of periodic functions of high smoothness. The simplest example of a linear approximation of continuous periodic functions of a real variable is the approximation by partial sums of the Fourier series. The sequences of partial Fourier sums are not uniformly convergent over the class of continuous periodic functions. A significant number of works is devoted to the study of other approximation methods, which are generated by transformations of Fourier sums and allow us to construct trigonometrical polynomials that would be uniformly convergent for each continuous function. Over the past decades, Fejer sums and de la Vallee Poussin sums have been widely studied. One of the most important direction in this field is the study of the asymptotic behavior of upper bounds of deviations of linear means of Fourier sums on different classes of periodic functions. Methods of investigation of integral representations of deviations of trigonometric polynomials generated by linear methods of summation of Fourier series, were originated and developed in the works of S.M. Nikolsky, S.B. Stechkin, N.P. Korneichuk, V.K. Dzadyk and others. The aim of the work systematizes known results related to the approximation of classes of Poisson integrals by arithmetic means of Fourier sums, and presents new facts obtained for particular cases. In the paper is studied the approximative properties of repeated Fejer sums on the classes of periodic analytic functions of real variable. Under certain conditions, we obtained asymptotic formulas for upper bounds of deviations of repeated Fejer sums on classes of Poisson integrals. The obtained formulas provide a solution of the corresponding Kolmogorov-Nikolsky problem without any additional conditions.

Some Tauberian conditions on logarithmic density

This article is based on the study on the λ-statistical convergence with respect to the logarithmic density and de la Vallee Poussin mean and generalizes some results of logarithmic λ-statistical convergence and logarithmic (V,lambda )-summability theorems. Hardy’s and Landau’s Tauberian theorems to the statistical convergence, which was introduced by Fast long back in 1951, have been extended by J.A. Fridy and M.K. Khan (Proc. Am. Math. Soc. 128:2347–2355, 2000) in recent years. In this article we try to generalize some Tauberian conditions on logarithmic statistical convergence and logarithmic (V,lambda )-statistical convergence, and we find some new results on it.

On approximations of the function |x|<sup>s</sup> by the Vallee Poussin means of the Fourier series by the system of the Chebyshev – Markov rational fractions

The approximative properties of the Valle Poussin means of the Fourier series by the system of the Chebyshev – Markov rational fractions in the approximation of the function |x|s, 0 < s < 2 are investigated. The introduction presents the main results of the previously known works on the Vallee Poussin means in the polynomial and rational cases, as well as on the known literature data on the approximations of functions with power singularity. The Valle Poussin means on the interval [–1,1] as a method of summing the Fourier series by one system of the Chebyshev – Markov rational fractions are introduced. In the main section of the article, a integral representation for the error of approximations by the rational Valle Poussin means of the function |x|s, 0 < s < 2, on the segment [–1,1], an estimate of deviations of the Valle Poussin means from the function |x|s, 0 < s < 2, depending on the position of the point on the segment, a uniform estimate of deviations on the segment [–1,1] and its asymptotic expression are found. The optimal value of the parameter is obtained, at which the deviation error of the Valle Poussin means from the function |x|s, 0 < s <2, on the interval [–1,1] has the highest velocity of zero. As a consequence of the obtained results, the problem of approximation of the function |x|s, s > 0, by the Valle Poussin means of the Fourier series by the system of the Chebyshev first-kind polynomials is studied in detail. The pointwise estimation of approximation and asymptotic estimation are established.The work is both theoretical and applied. Its results can be used to read special courses at mathematical faculties and to solve specific problems of computational mathematics.

Sums where fractional parts appear

In this article we examine sums where fractional parts appear. These sums are a generalization of the well-known De la Vallee Poussin's formula.

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.

Приближение аналитических функций повторными суммами Валле Пуссена

Приближение аналитических функций повторными суммами Валле Пуссена

Approximation Properties of Repeated de la Vallée-Poussin Means for Piecewise Smooth Functions

Basing on Fourier’s trigonometric sums and the classical de la Vallee-Poussin means, we introduce the repeated de la Vallee-Poussin means. Under study are the approximation properties of the repeated means for piecewise smooth functions. We prove that the repeated means achieve the rate of approximation for the discontinuous piecewise smooth functions which is one or two order higher than the classical de la Vallee-Poussin means and the partial Fourier sums respectively.

Integral presentation of deviations of rectangular linear means of Fourier series on classes of periodic differentiable functions

The paper deals with the problems of approximation in a uniform metric of periodic functions of many variables by trigonometric polynomials, which are generated by linear methods of summation of Fourier series. Questions of asymptotic behavior of the upper bounds of deviations of linear operators generated by the use of linear methods of summation of Fourier series on the classes of periodic differentiable functions are studied in many works. Methods of investigation of integral representations of deviations of polynomials on the classes of periodic differentiable functions of real variable originated and received its development through the works of S.M. Nikol'skii, S.B. Stechkin, N.P.Korneichuk, V.K. Dzadik, A.I. Stepanets, etc. Along with the study of approximation by linear methods of classes of functions of one variable, are studied similar problems of approximation by linear methods of classes of functions of many variables. In addition to the approximative properties of rectangular Fourier sums, are studied approximative properties of other approximation methods: the rectangular sums of Valle Poussin, Zigmund, Rogozinsky, Favar. In this paper we consider the classes of \(\overline{\psi}\)-differentiable periodic functions of many variables, allowing separately to take into account the properties of partial and mixed \(\overline{\psi}\)-derivatives, and given by analogy with the classes of \(\overline{\psi}\)-differentiable periodic functions of one variable. Integral representations of rectangular linear means of Fourier series on classes of \(\overline{\psi}\)-differentiable periodic functions of many variables are obtained. The obtained formulas can be useful for further investigation of the approximative properties of various linear rectangular methods on the classes \(\overline{\psi}\)-differentiable periodic functions of many variables in order to obtain a solution to the corresponding Kolmogorov-Nikolsky problems.

Criteria for the Best Approximation by Simple Partial Fractions on Semi-Axis and Axis

We study uniform approximation of real-valued functions f, f(∞) = 0, on ℝ+ and ℝ by real-valued simple partial fractions (the logarithmic derivatives of polynomials). We obtain a criterion for the best approximation on ℝ+ and ℝ in terms of the Chebyshev alternance. This criterion is similar to the known criterion on finite segments. For the problem of approximating odd functions on ℝ we construct an alternance criterion with a weakened condition on the poles of fractions. We present a criterion for the best approximation by simple partial fractions on ℝ+ and ℝ in terms of Kolmogorov. We prove analogs of the de la Vallee-Poussin alternation theorem.

A de la Vallée Poussin Type Inequality on Time Scales

In this work we derive an inequality of the de la Vallee Poussin type on a general time scale. In particular, we believe that a discrete version of such inequality is presented for the first time in the literature.

On the norms and minimal properties of de la Vallée Poussin’s type operators

In this paper we consider the de la Vallee Poussin’s type operators \(H_{rn,sn}\) $$\begin{aligned} H_{rn,sn}:=\frac{F_{rn}+F_{rn+1}+...+F_{sn-1}}{(s-r)n}, \end{aligned}$$ where \(F_k\) are classical Fourier projections onto \(\varPi _k\) (the space of trigonometric polynomials of degree less than or equal to k). We determine when \(H_{n,sn}\) is the minimal generalized projection and provide the asymptotic behavior of the norm \(\Vert H_{n,sn}\Vert \). Additionally, we contrast the results obtained for the trigonometric system to the results obtained for the Rademacher system.