On mean convergence of quasi-Hermite-interpolation

In this paper, the mean convergence of the quasi-Hermite-Fejér interpolation R_n(f,x) and the mean conergence of the quasi-Hermite interpolation R _n ^* (f,x) and its derivative based on the zeros of (1−x²)P_n (where P_n(x) be the Legendre polynomial) are respectively considered. The degree of corresponding L₂-approximation is given.     

Mean convergence of Hermite-Fejér interpolation

Weighted L^p convergence of Hermite-Fejér interpolation based on the zeros of generalized Jacobi polynomials is investigated. The main result of the paper gives necessary and sufficient conditions for such convergence for all continuous functions.     

Non-Commutative Carathéodory Interpolation

We prove a Carathéodory–Fejér type interpolation theorem for certain matrix convex sets in \mathbbC^d{\mathbb{C}^d} using the Blecher–Ruan–Sinclair characterization of abstract operator algebras. Our results generalize the work of Dmitry S. Kalyuzhnyĭ-Verbovetzkiĭ for the d-dimensional non-commutative polydisc.     

Korovkin-type convergence results for non-positive operators

Korovkin-type approximation theory usually deals with convergence analysis for sequences of positive operators. In this work we present qualitative Korovkin-type convergence results for a class of sequences of non-positive operators, more precisely regular operators with vanishing negative parts under a limiting process. Sequences of that type are called sequences of almost positive linear operators and have not been studied before in the context of Korovkin-type approximation theory. As an example we show that operators related to the multivariate scattered data interpolation technique moving least squares interpolation originally due to Lancaster and Šalkauskas [Surfaces generated by moving least squares methods, Math. Comp., 1981, 37, 141–158] give rise to such sequences. This work also generalizes Korovkin-type results regarding Shepard interpolation [Korovkin-type convergence results for multivariate Shepard formulae, Rev. Anal. Numér. Théor. Approx., 2009, 38, 170–176] due to the author. Moreover, this work establishes connections and differences between the concepts of sequences of almost positive linear operators and sequences of quasi-positive or convexity-monotone linear operators introduced and studied by Campiti in [Convexity-monotone operators in Korovkin theory, Rend. Circ. Mat. Palermo (2) Suppl., 1993, 33, 229–238].     

One-sided convergence conditions for Hermite-Fejér interpolation of higher order of Lagrange type

The authors investigate the Hermite-Fejér interpolation of higher order of Lagrange type for continuous functions on the Jacobi abscissas. A uniform convergence theorem is stated, generalizing a previous result for Lagrange interpolation.     

Error bounds for approximation in Chebyshev points

This paper improves error bounds for Gauss, Clenshaw–Curtis and Fejér’s first quadrature by using new error estimates for polynomial interpolation in Chebyshev points. We also derive convergence rates of Chebyshev interpolation polynomials of the first and second kind for numerical evaluation of highly oscillatory integrals. Preliminary numerical results show that the improved error bounds are reasonably sharp.     

ON THE SATURATION OF Lw^p-APPROXIMATIONBY （O-q＇-q） TYPE HERMITE-FEJER INTERPOLATING POLYNOMIALS

The “o” saturation theorem and the degree of L^p _w approximation by (0−q′−q) type Hermite-Fejér interpolating polynomials for mean convergence are obtained. This work is supported by the Doctor Foundation (No:02J20102-06) and the Post Doctor Foundation of Ningbo University.     

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.     

Weighted Fejér constants and Fekete sets

We indicate the connections among the Fekete set, the zeros of orthogonal polynomials, 1(w)-normal point systems, and the nodes of an interpolatory process which is called stable and the most economical, via the Fejér constants. Finally the convergence of a weighted Grünwald interpolation is proved.     

Hermite-Fejer type interpolation of higher order

The paper introduces Hermite-Fejér type (Hermite type) interpolation of higher order denoted by S _mn(f)(S* _mm(f)), and gives some basic properties including expression formulas, convergence relationship between S _mn(f) and H _mn(f) (Hermite-Fejér interpolation of higher order), and the saturation of S _mn(f). Supported by the Science Foundation of Shanxi Province for Returned Scholars.     

Bivariate Lagrange interpolation at the Padua points: the ideal theory approach

The Padua points are a family of points on the square [−1, 1]² given by explicit formulas that admits unique Lagrange interpolation by bivariate polynomials. Interpolation polynomials and cubature formulas based on the Padua points are studied from an ideal theoretic point of view, which leads to the discovery of a compact formula for the interpolation polynomials. The L ^p convergence of the interpolation polynomials is also studied. S. De Marchi and M. Vianello were supported by the “ex-60%” funds of the University of Padua and by the INdAM GNCS (Italian National Group for Scientific Computing). Y. Xu was partially supported by NSF Grant DMS-0604056.     

Hermite—Fejér Interpolation for Rational Systems

This paper considers Hermite—Fejér and Grünwald interpolation based on the zeros of the Chebyshev polynomials for the real rational system P _n (a ₁ , . . . , a _n ) with the nonreal poles in {a}ⁿ _k=1 C\[-1,1] paired by complex conjugation. This extends some well-known results of Fejér and Grünwald for the classical polynomial case. July 11, 1996. Dates revised: January 6, 1997 and July 30, 1997.     

ON LAGRANGE INTERPOLATION TO ｜x｜^α（1 〈α 〈 2） WITH EQUALLY SPACED NODES

S.M. Lozinskii proved the exact convergence rate at the zero of Lagrange interpolation polynomials to |x| based on equidistant nodes in [−1,1], In 2000, M. Rever generalized S.M. Lozinskii’s result to |x|^α(0≤α≤1). In this paper we will present the exact rate of convergence at the point zero for the interpolants of |x|^α(1<α<2).     

Lagrange插值和Hermite-Fejér插值在Wiener空间下的平均误差

在L_q-范数逼近的意义下,确定了基于Chebyshev多项式零点的Lagrange插值多项式列和Hermite-Fejér插值多项式列在Wiener空间下的p-平均误差的弱渐近阶.从我们的结果可以看出,当2≤q<∞,1≤p<∞时,基于第一类Chebyshev多项式零点的Lagrange插值多项式列和Hermite-Fejér插值多项式列的p-平均误差弱等价于相应的最佳逼近多项式列的p-平均误差.在信息基计算复杂性的意义下,如果可允许信息泛函为计算函数在固定点的值,那么当1≤p,q<∞时,基于第一类Chebyshev多项式零点的Lagrange插值多项式列和Hermite-Fejér插值多项式列在Wiener空间下的p-平均误差弱等价于相应的最小非自适应p-平均信息半径.     

THE EXACT CONVERGENCE RATE AT ZERO OF LAGRANGE INTERPOLATION POLYNOMIAL TO ｜x｜^α

In this paper we present a generalized quantitative version of a result the exact convergence rate at zero of Lagrange interpolation polynomial to spaced nodes in [-1,1] due to M.Revers concerning f（x） = ｜x｜α with on equally     

A geometric feature for finite element schemes

In this note we characterize the geometric feature of a (μ;r,k)—FES. Namely, for a C^μ triangular interpolation scheme with C^r vertex data, any angle of the macrotriangle must be divided into at least (μ+1)/(r+1−μ) parts.     

On summability of weighted Lagrange interpolation. III

Starting from the Lagrange interpolation on the roots of Jacobi polynomials, a wide class of discrete linear processes is constructed using summations. Some special cases are also considered, such as the Fejér, de la Vallée Poussin, Cesàro, Riesz and Rogosinski summations. The aim of this note is to show that the sequences of this type of polynomials are uniformly convergent on the whole interval [-1,1] in suitable weighted spaces of continuous functions. Order of convergence will also be investigated. Some statements of this paper can be obtained as corollaries of our general results proved in [15].     

On the phenomenon of spurious interpolation of elliptic functions by diagonal Padé approximants

We study diagonal Padé approximants for elliptic functions. The presence of spurious poles in the approximants not corresponding to the singularities of the original function prevents uniform convergence of the approximants in the Stahl domain. This phenomenon turns out to be closely related to the existence in the Stahl domain of points of spurious interpolation at which the Padé approximants interpolate the other branch of the elliptic function. We also investigate the behavior of diagonal Padé approximants in a neighborhood of points of spurious interpolation.     

On the global central limit theorem for ϕ-mixing random variables

The estimate of the remainder term is obtained in the global central limit theorem for π-mixing r.v.s. As a consequence of Theorem 1 the convergence rate of absolute moments for sums of π-mixing r.v.s. to corresponding absolute moments of the normal r.v. is found. Published in Lietuvos Matematikos Rinkinys, Vol. 35, No. 2, pp. 233–247, April–June, 1995.