Error Estimates and Reproduction of Polynomials for Biorthogonal Local Trigonometric Bases

A general approach for biorthogonal local trigonometric bases in the two-overlapping setting was given by Chui and Shi. In this paper, we give error estimates for the approximation with such basis functions. In particular, it is shown that for a partition of the real axis into small intervals one obtains better approximation order if polynomials are reproduced locally. Furthermore, smooth trigonometric bases are constructed, which reproduce constants resp. linear functions by only one resp. a small number of basis functions for each interval.     

Approximation of Linear Age-Structured Population Models Using Legendre Polynomials

We develop a numerical algorithm for approximation of solutions for linear age-structured population models. The construction is based on approximation of age distributions by modified Legendre polynomials and uses the Trotter-Kato theorem of semigroup theory for the corresponding abstract Cauchy problem. Unbounded resp. nonintegrable mortality rates are admissible.     

Approximation by Homogeneous Polynomials

Let be the boundary of a convex domain symmetric to the origin. The conjecture that any continuous even function can be uniformly approximated by homogeneous polynomials of even degree on K is proven in the following cases: (a) if d = 2; (b) if K is twice continuously differentiable and has positive curvature in every point; or (c) if K is the boundary of a convex polytope.     

Approximation by Algebraic Polynomials on Rectangles

A direct theorem for approximation by algebraic polynomials in two variables with different degrees in each variable in L_p-metric (1 p ) on rectangles is proved, and the dependence of the constants on various parameters is studied.     

Wilson Bases and Modulation Spaces

We show that the recently discovered WILSON bases of exponential decay are unconditional bases for all modulation spaces on R, including the classical BESSEL potential spaces, the Segal algebra S_o, and the SCHWARTZ space. As a consequence we obtain new bases for spaces of entire functions. On the other hand, the WILSON bases are no unconditional bases for the ordinary L^p-spaces for p ≠ 2.     

Taylor多项式逼近函数的计算机模拟

针对典型教学案例,借助MATLAB的绘图与动画功能,通过编制程序并绘制图像,探讨Taylor多项式逼近函数的直观解释,以求帮助学生深入理解函数的Taylor展开式.     

Cωp空间中的插值多项式逼近

本文证明了对任意函数f(z)∈C_ω^p.     

On Simultaneous Approximation by Optimal Algebraic Polynomials

In this paper the order of bestapproximation by algebraic polynomials is related to the order of weighted simultaneous approximation. A special case of a Markov-Bernstein type inequality proved by Ditzian and Totik in a very general fashion is used.     

Integral Representations of the Wilson Polynomials and the Continuous Dual Hahn Polynomials

We give representations of the Wilson polynomials and the continuous dual Hahn polynomials in terms of multidimensional generalizations of Barnes type integrals. Motivation is to study the Barnes type integrals from the viewpoint of the de Rham theory and holonomic systems.     

The Bernoulli Spline and Approximation by Trigonometric Blending Polynomials

Using an extension of the notion of Bernoulli spline to a multivariate setting, a Jackson-Favard estimate is derived for approximation of 2π-periodic test functions by trigonometric blending polynomials. Techniques involved in the proof include properties of periodic distributions and of their Fourier transforms. The usual Jackson-Favard estimates from the literature can be derived from our result if limits of test functions are considered.     

Low-Degree Approximation of Random Polynomials

Foundations of Computational Mathematics - We prove that with “high probability” a random Kostlan polynomial in $$n+1$$ many variables and of degree d can be approximated by a...     

Approximation Properties of Julia Polynomials

Let G be a finite simply connected domain in the complex plane C, bounded by a rectifiable Jordan curve L, and let w = φ0 （z） be the Riemann conformal mapping of G onto D （0, r0） ：= {E-mail： ： ｜｜ 〈 r0}, normalized by the conditions φ0 （z0） = 0, φ＇0 （z0） = 1. In this work, the rate of approximation of φ0 by the polynomials, defined with the help of the solutions of some extremal problem, in a closed domain G is studied. This rate depends on the geometric properties of the boundary L.     

Approximation by Complex Bernstein-Durrmeyer Polynomials in Compact Disks

In this paper, the order of simultaneous approximation and Voronovskaja kind results with quantitative estimate for complex Bernstein-Durrmeyer polynomials attached to analytic functions on compact disks are obtained. In this way, we put in evidence the overconvergence phenomenon for Bernstein-Durrmeyer polynomials, namely the extensions of approximation properties (with quantitative estimates) from real intervals to compact disks in the complex plane.     

Approximation by Vilenkin Polynomials in Weighted Orlicz Spaces

Analysis Mathematica - Direct and inverse theorems are proved for approximation by bounded Vilenkin systems in weighted Orlicz spaces. The equivalence between the K-functional constructed with help...     

WKB Approximation and Krall-Type Orthogonal Polynomials

We give a unified approach to the Krall-type polynomials orthogonal withrespect to a positive measure consisting of an absolutely continuous oneperturbed by the addition of one or more Dirac deltafunctions. Some examples studied by different authors are considered from aunique point of view. Also some properties of the Krall-type polynomials arestudied. The three-term recurrence relation is calculated explicitly, aswell as some asymptotic formulas. With special emphasis will be consideredthe second order differential equations that such polynomials satisfy. Theyallow us to obtain the central moments and the WKB approximation of thedistribution of zeros. Some examples coming from quadratic polynomialmappings and tridiagonal periodic matrices are also studied.     

有限Diophantine逼近与Fibonacci多项式

设x为一无理数具简单的连分式展开x = [a0, a1, a2,…, an].若对无穷多个是标k有ak> n则至少有m个解p/q(p与q互素)使不等式 x - p/q 相似文献   

Approximation of Periodic Analytic Functions by Interpolation Trigonometric Polynomials

We obtain asymptotic equalities for the upper bounds of approximations by interpolation trigonometric polynomials on classes of convolutions of periodic functions admitting a regular extension to a fixed strip of the complex plane.     

On Approximation by Reciprocals of Polynomials with Positive Coefficients

In order to study the approximation by reciprocals of polynomials with real coefficients, one always assumes that the approximated function has a fixed sign on the given interval. Sometimes, the approximated function is permitted to have finite sign changes, such as $l(l\geq1)$ times. Zhou Songping has studied the case $l=1$ and $l\geq2$ in $L^{p}$ spaces in order of priority. In this paper, we studied the case $l\geq2$ in Orlicz spaces by using the function extend, modified Jackson kernel, Hardy-Littlewood maximal function, Cauchy-Schwarz inequality, and obtained the Jackson type estimation.