Bergman空间的插值多项式逼近

本文在1相似文献   

渐近Fejer点上的Lagrange插值多项式的逼近阶

本文考虑渐近 Fejer 点上 Lagrange 插值多项式在 Jordan 区域 D 边界上一致逼近及平均逼近 A(D)中的函数,得到了逼近阶的估计式。     

复曲线上的Jackson型定理

令$\Ga$是复平面(z)中的光滑闭Jordan曲线. 作者借助于Hermite插值的基多项式, 引入连续函数插值, 它一致收敛于$f(z)\in C(\Ga)$,且具有和实区间$[-1, 1]$上Jackson定理1中一样的逼近阶, 并证明了这里逼近阶的精确性. 利用和以往工作不同的方法, 研究了同时逼近到函数及其导数, 并得到和实区间$[-1, 1]$上Jackson定理2一样的理想结果.     

一类有理Bézier曲线及其求积求导的多项式逼近

用多项式曲线来逼近有理曲线在计算机辅助几何设计(CAGD)系统中可简化求积求导等繁琐的计算.然而,按现有的方法能检验一条已知的有理曲线是否具有收敛的多项式逼近曲线却不易选择适当的权因子来产生能用多项式曲线来加以逼近的有理曲线,即不易做到事先设计;同时,要减少求积、求导的逼近误差只能依靠提高多项式曲线的次数.文中给出一类有理Bézier曲线及其多项式逼近算法较好地克服了这两种缺陷,具有推广应用的价值.     

单纯形上的Bernstein多项式

本文研究了单纯形上的Bernstein多项式的一系列性质.我们给出了Bernstein多项式逼近连续函数的精确误差界,确定了Bernstein多项式的最佳逼近度,并得到了Bernstein算子及其逆算子的渐近展开式.最后,这些结果被应用于单纯形上Bezier网的研究.     

多元Stancu多项式与连续模

本文研究单纯形上多元Stancu多项式与连续模之间的关系,证明了Stancu多项式具有保持连续模的性质,推广了一元Bernstein多项式的相应结果.同时,利用多元函数的Ditzian-Totik连续模估计Stancu多项式逼近多元连续函数速度的上界和下界,得到一个使得逼近速度为O(n-a)(0相似文献   

复平面上亚纯函数的有理函数插值逼近

在本文中考虑光滑度为1+ε的Jordan区域D,ε>0,得到了在区域内具有有限个极点的亚纯函数在Fejer插值基点上的有理插值函数的L^p(?D)空间中的平均逼近阶,0相似文献   

关于矩阵多项式Jordan标准形刻画

本文利用矩阵运算、矩阵相似关系及矩阵的秩,深化了Jordan矩阵的性质,并在此基础上刻画了矩阵Jordan标准形中Jordan块的个数及阶数,最后讨论了矩阵多项式Jordan标准形,充实了高等代数中Jordan标准形的结果．     

关于苏联科学院数学研究所在函数逼近论方面的工作（上）

本文全面介绍了苏联科学院数学研究所的逼近论专家们多年来在函数逼近论的各个领域内所做的工作，这些领域包括：多项式的极值性质，一元周期函数与非周期函数逼近的正逆定理，线性逼近方法，多元函数逼近，函数类的宽度和求积公式等，在叙述这些成果的同时，作者还介绍了前人的有关结果．     

扰动Fej\'{e}r点上复Hermite插值的联合逼近

$D$是复平面中由闭Jordan曲线$\Ga$围成的单连区域. 考虑在$\Ga$上扰动Fej\''er点的 Hermite插值一致逼近、平均逼近和联合逼近于函数$f\in A^{(q)}(\o D)$. 该文中的逼近阶一般说来是不可再改进的, 区域的边界限制条件到目前为止是最少的. 以往的全部同类结果都包括在该文中作为特殊情形, 由于该文方法上的改进, 简化和省去了以往 某些证明过程.     

The distribution of zeros of asymptotically extremal polynomials

In this paper, we study the asymptotic behavior of the zeros of a sequence of polynomials whose weighted norms have the same nth root behavior as the weighted norms for certain extremal polynomials. Our results include as special cases several of the previous results of Erd s, Freud, Jentzsch, Szeg and Blatt, Saff, and Simkani. Applications are given concerning the zeros of orthogonal polynomials over a smooth Jordan curve (in particular, on the unit circle) and the zeros of polynomials of best approximation on R to nonentire functions.     

Erdös-Turán theorems on a system of Jordan curves and arcs

Erdös and Turán established in [4] a qualitative result on the distribution of the zeros of a monic polynomial, the norm of which is known on [–1, 1]. We extend this result to a polynomial bounded on a systemE of Jordan curves and arcs. If all zeros of the polynomial are real, the estimates are independent of the number of components ofE for any regular compact subsetE ofR. As applications, estimates for the distribution of the zeros of the polynomials of best uniform approximation and for the extremal points of the optimal error curve (generalizations of Kadec's theorem) are given.Communicated by Dieter Gaier.     

Best one-sided L1 approximation to the Heaviside and sign functions

We find the polynomials of the best one-sided approximation to the Heaviside and sign functions. The polynomials are obtained by Hermite interpolation at the zeros of some Jacobi polynomials. Also we give an estimate of the error of approximation and characterize the extremal points of the convex set of the best approximants.     

An Extremal Relation of the Theory of Approximation of Functions by Algebraic Polynomials

In this paper we find an extremal relation of the theory of approximation of functions on a finite interval by algebraic polynomials. In the Legendre case we find some estimations and representation in the form of a series of the constant from the extremal relation.     

Smooth equilibrium measures and approximation

A necessary and sufficient condition is given for approximation with weighted expressions of the form wnPn, where w is a given continuous weight function and Pn are polynomials of degree . The condition is that the extremal measure that solves an associated equilibrium problem is smooth (asymptotically optimal doubling). As corollaries we get all previous (positive and negative) results for approximation, as well as the solution of a problem of T. Bloom and M. Branker. A connection to level curves of homogeneous polynomials of two variables is also explored.     

Zur approximation des transfiniten durchmessers bei bis auf ecken analytischen geschlossenen jordankurven

LetC be a closed Jordan curve in the complex plane and letf(z)=dz+a ₀+a ₁ z ^?1+… be the analytic function mapping |z|>1 schlicht onto the exterior ofC (d>0 is the transfinite diameter ofC). Similar to the Fekete points a point system will be defined calledextremal points. One can use the Fekete points or the extremal points to approximated. The author has proved [3] that in the case of an analytic closed Jordan curve the approximation ofd by means of extremal points is much better than the approximation ofd by the use of Fekete points. Here we show how to approximated by means of extremal points in the case of a piecewise analytic, closed Jordan curve possessing corners of openingαπ (0<α<2).     

Extremal problems for linear functionals on the Tchebycheff spaces

The study of Tchebycheff spaces (generalizing the space of algebraic polynomials) and extremal problems related to them began one and a half centuries ago. Recently, many facts of approximation theory have been understood and reinterpreted from the point of view of general principles of the theory of extremum and convex duality. This approach not only allowed one to prove the previously known results for algebraic polynomials and generalized polynomials in a unified way, but also enabled one to obtain new results. In this paper, we work out this direction with special attention to the optimal recovery problems. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 2, pp. 87–100, 2005.     

Invariants and Chebyshev polynomials

On different compact sets from ℝ ⁿ , new multidimensional analogs of algebraic polynomials least deviating from zero (Chebyshev polynomials) are constructed. A brief review of the analogs constructed earlier is given. Estimates of values of the best approximation obtained by using extremal signatures, lattices, and finite groups are presented.     

Invariants and Chebyshev polynomials

On different compact sets from ? ⁿ , new multidimensional analogs of algebraic polynomials least deviating from zero (Chebyshev polynomials) are constructed. A brief review of the analogs constructed earlier is given. Estimates of values of the best approximation obtained by using extremal signatures, lattices, and finite groups are presented.     

Extremal properties of quadratic differentials with strip-shaped domains in the structure of the trajectories

One considers the modulus problem for a family of homotopic classes {H_i}, in the extended complex plane, of the following types. The classes H_n consist of closed Jordan curves, homotopic to appropriate nondegenerate contours or point curves, and also of arcs with endpoints in (distinct or coinciding) distinguished points in. One establishes the relation of the indicated extremal metric problem and the problem on the extremal decomposition of in the family of systems of mutually nonoverlapping domains {D_i}, associated with the family of the classes {H_i}. The results of this paper complement a previous theorem of the author [Moduli of families of curves and quadratic differentials. Trudy Mat. Inst. Akad. Nauk SSSR, Vol. 139, 1980].Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 154, pp. 110–129, 1986.