A new modification of the Hermite-Fejér interpolation

пУсть жАДАНы Ужлы $$ - \infty< x_1< x_2< ...< x_k< x_{k + 1}< ...< x_n< + \infty ,$$ , И пУстьx ₁ ^* <x ₂ ^* <...<x _n-1 ^* — кОРНИ МНОгО ЧлЕНА Ω′(х). гДЕ $$\omega (x) = \prod\limits_{k = 1}^n {(x - x_k ).} $$ В РАБОтЕ ИсслЕДУЕтсь жАДАЧА: кАк ОпРЕДЕлИт ь МНОгОЧлЕНР(х) МИНИМАльНОИ стЕп ЕНИ, Дль кОтОРОгО ВыпОлНь Утсь слЕДУУЩИЕ ИНтЕР пОльцИОННыЕ УслОВИь гДЕ {y _k И {y _k′}-жАДАННы Е сИстЕМы жНАЧЕНИИ.     

Hermite-Fejér interpolation of higher order with varying weights

The authors obtain precise estimations for the coefficients of Hermite-Fejér interpolation of higher order based on Generalized Jacobi zeros.     

Asymptotic constants for the error of Hermite-Fejér interpolation on the unit circle

The paper deals with the rate of convergence for the Laurent polynomials of Hermite-Fejér interpolation on the unit circle with nodal system the n roots of a complex number with modulus one. The order of convergence and the asymptotic constants are obtained when we consider analytic functions on open disks and open annulus containing the unit circle.     

On the Rough Theory of Hermite-Fejér Type Interpolation of Higher Order

This paper proves that there does not exist the rough theory for Hermite-Fejér type interpolation of higher order.     

The Exact Estimation of the Hermite-Fejér Interpolation

The exact pointwise estimation of the Hermite-Fejér interpolation process based on the zeros of the Jacobi polynomial $P^{(\alpha,\beta)}_n(x)(-1 ‹\alpha,\beta \leq 0)$ is given. The method employed is useful for other extended H-F interpolations also.     

On the divergence of Walsh-Fejér means of bounded functions on sets of measure zero

The aim of this paper is to prove that for an arbitrary set of measure zero there exists a bounded function for which the Fejér means of the Walsh-Fourier series of the function diverge. Research supported by the Georgian National Fundation for Scientific Research, Grant no. 07_225_3-100.     

The average errors for Hermite-Fejr interpolation on the Wiener space

For 1≤ p ∞, firstly we prove that for an arbitrary set of distinct nodes in [-1, 1], it is impossible that the errors of the Hermite-Fejr interpolation approximation in L p -norm are weakly equivalent to the corresponding errors of the best polynomial approximation for all continuous functions on [-1, 1]. Secondly, on the ground of probability theory, we discuss the p-average errors of Hermite-Fejr interpolation sequence based on the extended Chebyshev nodes of the second kind on the Wiener space. By our results we know that for 1≤ p ∞ and 2≤ q ∞, the p-average errors of Hermite-Fejr interpolation approximation sequence based on the extended Chebyshev nodes of the second kind are weakly equivalent to the p-average errors of the corresponding best polynomial approximation sequence for L q -norm approximation. In comparison with these results, we discuss the p-average errors of Hermite-Fejr interpolation approximation sequence based on the Chebyshev nodes of the second kind and the p-average errors of the well-known Bernstein polynomial approximation sequence on the Wiener space.     

On the Hermite interpolation

Explicit representations for the Hermite interpolation and their derivatives of any order are provided.Furthermore,suppose that the interpolated function f has continuous derivatives of sufficiently high order on some sufficiently small neighborhood of a given point x and any group of nodes are also given on the neighborhood.If the derivatives of any order of the Hermite interpolation polynomial of f at the point x are applied to approximating the corresponding derivatives of the function f(x),the asymptotic representations for the remainder are presented.     

关于Hermite-Fejér型算子

§1.引言 设ω(t)是给定的连续模,H_ω={f;ω(f,t)≤ω(t)}。P_n~(α,β)(x)(α,β>-1)表示n阶Jacobi多项式;P_n(x)=P_n~(0,0)(x)为Legendre多项式。 定义1 (见[1,555页])设{x_κ~((n))}_(κ=1)~n(n=1,2,…)为属于区间[-1,1]的节点系。     

On the maximal operator of Walsh-Kaczmarz-Fejér means

In this paper we prove that the maximal operator      

On the maximal operators of Walsh-Kaczmarz-Fejér means

The main aim of this paper is to prove that the maximal operator $\sigma _p^{\kappa , * } f: = \sup _{n \in P} {{\left| {\sigma _n^\kappa f} \right|} \mathord{\left/ {\vphantom {{\left| {\sigma _n^\kappa f} \right|} {\left( {n + 1} \right)^{{1 \mathord{\left/ {\vphantom {1 {p - 2}}} \right. \kern-0em} {p - 2}}} }}} \right. \kern-0em} {\left( {n + 1} \right)^{{1 \mathord{\left/ {\vphantom {1 {p - 2}}} \right. \kern-0em} {p - 2}}} }}$ is bounded from the Hardy space H _p to the space L _p for 0 < p < 1/2.     

On the divergence of de la Vallée Poussin means of Fourier series

Пусть {λ _{n 1} ^t8 — монотонн ая последовательнос ть натуральных чисел. Дл я каждой функции fεL(0, 2π) с рядом Фурье строятся обобщенные средние Bалле Пуссена $$V_n^{(\lambda )} (f;x) = \frac{{a_0 }}{2} + \mathop \sum \limits_{k = 1}^n (a_k \cos kx + b_k \sin kx) + \mathop \sum \limits_{k = n + 1}^{n + \lambda _n } \left( {1 - \frac{{k - n}}{{\lambda _n + 1}}} \right)\left( {a_k \cos kx + b_k \sin kx} \right).$$ Доказываются следую щие теоремы.

1. Если λ_n=o(n), то существуе т функция fεL(0, 2π), для кот орой последовательность {V_n ^(λ)(?;x)} расходится почти вс юду.
2. Если λ_n=o(n), то существуе т функция fεL(0, 2π), для кот орой последовательность $$\left\{ {\frac{1}{\pi }\mathop \smallint \limits_{ - \pi /\lambda _n }^{\pi /\lambda _n } f(x + t)\frac{{\sin (n + \tfrac{1}{2})t}}{{2\sin \tfrac{1}{2}t}}dt} \right\}$$ расходится почти всю ду

.     

高阶Hermite-Fejér型内插

The paper introduces Hermite-Fejr type (Hermite type) interpolation of higher order denoted by Smn(f)(S*mn(f)), and gives some basic properties including expression formulas, convergence relationship between Smn(f) and Hmn(f) (Hermite-Fejr interpolation of higher order), and the saturation of Smn(f).     

高阶Hermite-Fejér型插值理论

本文建立了高阶 Hermite-Fejér 型插值理论,该理论包括收敛准则、误差的下界估计及饱和性等.     

On the trigonometric polynomials of Fejér and Young

The trigonometric polynomials of Fejér and Young are defined by $S_n (x) = \sum\nolimits_{k = 1}^n {\tfrac{{\sin (kx)}} {k}}$S_n (x) = \sum\nolimits_{k = 1}^n {\tfrac{{\sin (kx)}} {k}} and $C_n (x) = 1 + \sum\nolimits_{k = 1}^n {\tfrac{{\cos (kx)}} {k}}$C_n (x) = 1 + \sum\nolimits_{k = 1}^n {\tfrac{{\cos (kx)}} {k}}, respectively. We prove that the inequality $\left( {{1 \mathord{\left/ {\vphantom {1 9}} \right. \kern-\nulldelimiterspace} 9}} \right)\sqrt {15} \leqslant {{C_n \left( x \right)} \mathord{\left/ {\vphantom {{C_n \left( x \right)} {S_n \left( x \right)}}} \right. \kern-\nulldelimiterspace} {S_n \left( x \right)}}$\left( {{1 \mathord{\left/ {\vphantom {1 9}} \right. \kern-\nulldelimiterspace} 9}} \right)\sqrt {15} \leqslant {{C_n \left( x \right)} \mathord{\left/ {\vphantom {{C_n \left( x \right)} {S_n \left( x \right)}}} \right. \kern-\nulldelimiterspace} {S_n \left( x \right)}} holds for all n ≥ 2 and x ∈ (0, π). The lower bound is sharp.     

Hermite-Fejér插值的收敛准则

本文给出Hermite-Fejér插值的若干收敛准则.其中之一是:Hermite-Fejer插值算子对每一个连续函数一致收敛当且仅当该算子范数一致有界且该算子对两个单项式x及x²一致收敛.