Coincidence of Least Uniform Deviations of Functions from Polynomials and Rational Fractions

For a given nonincreasing vanishing sequence {a _n } _{n = 0} of nonnegative real numbers, we find necessary and sufficient conditions for a sequence {n _k } _{k = 0} to have the property that for this sequence there exists a function f continuous on the interval [0,1] and satisfying the condition that , k = 0,1,2,..., where E _n (f) and R _n,m (f) are the best uniform approximations to the function f by polynomials whose degree does not exceed n and by rational functions of the form r _n,m (x) = p _n (x)/q _m (x), respectively.     

Bergman空间B_q~p(p>0,q>1)中的多项式最佳逼近的逆定理

本文在Bergman空间Bqp(p>0,q>1)中得到了关于用多项式逼近该空间函数的最佳逼近误差的阶的估计的逆定理．     

Bergman空间B_q~p(p>0,q>1)中的多项式最佳逼近的逆定理

本文在Bergman空间Bqp(p>0,q>1)中得到了关于用多项式逼近该空间函数的最佳逼近误差的阶的估计的逆定理．     

The Order of Best Approximation in Some Classes of Functions

Starting from the equivalence between the Ditzian–Totik modulus and , where , in this article large classes of functions are introduced for which the modulus can be easily calculated. As a consequence, very good estimates for the bestapproximation are obtained. The attempts to estimate or calculate themodulus can be a very intricateproblem.     

The Discrete Relativistic Toda Molecule Equation and a Padé Approximation Algorithm

The relativistic Toda molecule equation (RTM) describes a one-parameter deformation of coefficients of the recurrence relation of a class of biorthogonal polynomials including the Szegö polynomials. In this paper, we present (i) explicit solutions of the discrete relativistic Toda molecule equation (d-RTM), (ii) a new Padé approximation algorithm for a given power series.     

一类连分数的有理逼近

设f(n)是非负函数,k,b,s_i,t_i(i=1,2,…)是正常数,研究形如[a_0,a_1,a_2…]=[■]_m~∞=0和[■]_n~∞=1的连分数有理逼近的下界．     

Approximation of functions of boundedp-variation by polynomials in terms of the faber-schauder system

In this paper the best polynomial approximation in terms of the system of Faber-Schauder functions in the spaceC _p [0, 1] is studied. The constant in the estimate of Jackson’s inequality for the best approximation in the metric ofC _p [0, 1] and the estimate of the modulus of continuity ω_1−1/p are refined. Translated fromMatematicheskie Zametki, Vol. 62, No. 3, pp. 363–371, September, 1997. Translated by N. K. Kulman     

Padé-Type Approximants of Markov and Meromorphic Functions

We study the question of convergence of Padé and Padé-type approximants to functions meromorphic in a domain. As an example we investigate in detail the case of functions of the formwhereμ(z) is a Markov function.     

Approximation by polynomials on quaternionic compact sets

In this paper, we obtain several extensions to the quaternionic setting of some results concerning the approximation by polynomials of functions continuous on a compact set and holomorphic in its interior. To this end, we prove an analog of the Riemann mapping theorem for a subclass of open sets, whose validity involves precisely the slice regular functions for which the composition remains slice regular. The results include approximation on compact starlike sets and compact axially symmetric sets. The cases of some concrete particular sets are described in details, including also quantitative estimates. Copyright © 2014 John Wiley & Sons, Ltd.     

On Approximation by Reciprocals of Spherical Harmonics in L p Norm

Let S^1-1,q≥2,be the surface of the unit sphere in the Euclidean space R^1,f（x）∈L^p（S^q-1）,f（x）≥0,f absohutely unegual to 0,1≤p≤＋∞,Then,it is proved in the present paper that there is a spherical harmonics PN（x） of order≤N and a constant C〉0 such that where ω（f,δ）L^p=sup 0〈t≤δ‖St（f）-f‖L^p is a kind of moduli of continuity and ^‖f-1/PN‖L^p≤Cω（f,N^-1）L^p,St（f,μ）=1/｜S^q-2｜Sin^2λt ∫-μμ’=t f（μ＇）dμ＇ is a translation operator.     

Interpolation of functions by the least squares method

We study the norm of a best quadratic trigonometric approximation operator on a finite uniform grid. Dedicated to the memory of my scientific adviser, Professor S. B. Stechkin Translated fromMatematicheskie Zametki, Vol. 66, No. 3, pp. 372–379, September, 1999.     

Orlicz空间中三角多项式倒数对周期可微函数的逼近定理

利用Orlicz空间内有关不等式技巧在Orlicz空间内研究了用三角多项式的倒数逼近周期可微函数的问题.得到了一个逼近定理及其推论.     

Computation of special functions by Padé approximants with orthogonal polynomial denominators

This paper deals with the computation of special functions of mathematics and physics in the complex domain using continued fraction (one-point or two-point Padé) approximants. We consider three families of continued fractions (Stieltjes fractions, real J-fractions and non-negative T-fractions) whose denominators are orthogonal polynomials or Laurent polynomials. Orthogonality of these denominators plays an important role in the analysis of errors due to numerical roundoff and truncation of infinite sequences of approximants. From the rigorous error bounds described one can determine the exact number of significant decimal digits contained in the approximation of a given function value. Results from computational experiments are given to illustrate the methods.Research supported in part by the National Science Foudation under Grant No. DMS-9302584.     

Cubic Thue inequalities with negative discriminant

We give an upper bound for the solutions of the family of cubic Thue inequalities |x³+axy²+by³|?k when a is positive and larger than a certain value depending on b. For the case k=a+|b|+1 and a?360b⁴ we show that these inequalities have only trivial solutions. For the case k=a+|b|+1 and |b|=1,2, we solve these inequalities for all a?1. Our method is based on Padé approximations using Rickert's integrals. We also use a generalization of Legendre's theorem on continued fractions.     

由斜坡函数激发的神经网络算子逼近

首先,引入一种由斜坡函数激发的神经网络算子,建立了其对连续函数逼近的正、逆定理,给出了其本质逼近阶.其次,引入这种神经网络算子的线性组合以提高逼近阶,并且研究了这种组合的同时逼近问题.最后,利用Steklov函数构造了一种新的神经网络算子,建立了其在L~p[a,b]空间逼近的正、逆定理.     

一种修正的拟Grünwald插值在Orlicz空间内的逼近度

修正了以第二类Chebyshev多项式的零点为插值结点组的拟Grünwald插值多项式,使之转化为积分形式,并利用不等式技巧和Hardy-Littlewood极大函数的方法,研究了此积分型拟Grünwald插值算子在带权Orlicz空间内的逼近问题,得出了意义相对广泛的逼近度估计的结果.     

Jackson type theorems on complex curves

Let Γ be a closed smooth Jordan curve in the complex plane. In this paper, with the help of a class of fundamental functions of Hermite interpolation, the author introduces a continuous function interpolation which uniformly approximates to f(z) ε C(Γ) with the same order of approximation as that in Jackson Theorem 1 on real interval [−1, 1]. The accuracy of the order of approximation is proved. Using the method different from the early works, the author studies simultaneous approximation to function and its derivatives and the desired results analogues to that in Jackson Theorem 2 on real interval [−1, 1] are obtained.      

Estimates for the Lebesgue functions and the Nevai formula for the sinc-approximations of continuous functions on an interval

We obtain some upper and lower estimates for the sequences of the Lebesgue functions and constants of the Whittaker operators

$L_n (f,x) = \sum\limits_{k = 0}^n {\frac{{\sin (nx - k\pi )}}{{nx - k\pi }}} f\left( {\frac{{k\pi }}{n}} \right)$

for continuous functions. We give an analog of Nevai’s formula for the Lagrange-Chebyshev and Lagrange-Laguerre interpolation polynomials for the operators under consideration. Its “local” version is established.

     

Approximation by polynomials in Bergman spaces of slice regular functions in the unit ball

In this paper, we show that the set of quaternionic polynomials is dense in the Bergman spaces of slice regular functions in the unit ball, both of the first and of the second kind. Several proofs are presented, including constructive methods based on the Taylor expansion and on the convolution polynomials. In the last case, quantitative estimates in terms of higher‐order moduli of smoothness and of best approximation quantity are obtained.     

Approximation of semigroups and cosine functions in spaces of periodic functions

In this article we furnish a representation of the solutions of some classes of first-order and second-order evolution problems as limit of iterates of classical sequences of approximating operators. The method is based on Trotter's theorem on the approximation of semigroups which is applied here also for the approximation of groups and cosine functions. We apply this method in spaces of continuous periodic functions and using some classical sequences of trigonometric polynomials.