Copositive approximation by rational functions with prescribed numerator degree

The paper proves that, if f（x） ∈ L^p[-1,1],1≤p〈∞ ,changes sign I times in （-1, 1）,then there exists a real rational function r（x） ∈ Rn^（2μ-1）l which is eopositive with f（x）, such that the following Jackson type estimate ｜｜f-r｜｜p≤Cδl^2μωφ（f,1/n）p holds, where μ is a natural number ≥3/2＋1/p, and Cδ is a positive constant depending only on δ.     

Cyclic AFD algorithm for the best rational approximation

We propose a practical algorithm of best rational approximation of a given order to a function in the Hardy H² space on the unit circle and on the real line. The type of approximation is proved to be equivalent with Blaschke form approximation. The algorithm is called Cyclic adaptive Fourier decomposition as it adaptively selects one parameter for each cycle on the basis of the maximal selection principle proved in the literature of adaptive Fourier decomposition. Copyright © 2013 John Wiley & Sons, Ltd.     

On meromorphic approximation

Let G be a bounded N-connected domain, the boundary of which consists of closed analytic Jordan curves. We assume that 0G. For any nonnegative integers n and m, denote by _n,m the class of all meromorphic functions on G that can be represented in the form h=p/qz ^m , where p belongs to the Smirnov class E (G), q is a polynomial of degree at most n, q0. Theorems giving necessary and sufficient conditions for a function belonging to the class _n,m to be an element of best approximation to a continuous function f on in the space L () by functions in the class _n,m are proved. Some questions concerning orthogonal polynomials and the theory of Hankel operators are also considered.     

On computing best Chebyshev complex rational approximants

An algorithm for computing best complex ordinary rational functions is presented. The final step of the procedure consists of solving the system of nonlinear equations defined by the local Kolmogorov criterion before checking recently developed sufficient optimality and uniqueness conditions. Various numerical results are reported exhibiting, in particular, nonunique solutions, saddle points and locally best approximants that are not global.     

The asymptotic property of approximation to |x| by Newman's rational operators

Let . The present note gives the asymptotoc formula of max . This revised version was published online in June 2006 with corrections to the Cover Date.     

On approximation by rational functions with prescribed numerator degree in L p spaces

Summary It is proved that, if <InlineEquation ID=IE"3"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"4"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"5"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"6"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"7"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"8"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"9"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"10"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>f(x)\in L^p_{[-1,1]}$, $1< p\ki \infty$, changes sign exactly $l$ times, then there exists a real rational function $r(x)\in R_{n}^l$ such that <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[$$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation> {\|f-r\|}_{p}\le C_{p,\delta}{(l+1)}^2\omega {(f,n^{-1})}_p, $$ which generalizes a result of Leviatan and Lubinsky in \cite{4}. A weaker similar result in $L^1_{[-1,1]}$ is also established.     

A method for rational Chebyshev approximation of rational functions on the unit disk and on the unit interval

A method for construction of CF approximants in some cases of rational approximation of a rational function f on the unit disk and on the unit interval is presented. The inverted square root of the greatest positive eigenvalue and a corresponding eigenvector of an eigenvalue problem defined by the coefficients of f gives the solution.     

On 3-monotone approximation by piecewise polynomials

We consider 3-monotone approximation by piecewise polynomials with prescribed knots. A general theorem is proved, which reduces the problem of 3-monotone uniform approximation of a 3-monotone function, to convex local L₁ approximation of the derivative of the function. As the corollary we obtain Jackson-type estimates on the degree of 3-monotone approximation by piecewise polynomials with prescribed knots. Such estimates are well known for monotone and convex approximation, and to the contrary, they in general are not valid for higher orders of monotonicity. Also we show that any convex piecewise polynomial can be modified to be, in addition, interpolatory, while still preserving the degree of the uniform approximation. Alternatively, we show that we may smooth the approximating piecewise polynomials to be twice continuously differentiable, while still being 3-monotone and still keeping the same degree of approximation.     

On the best linear approximation methods and the widths of certain classes of analytic functions

We discuss the best linear approximation methods in the Hardy spaceH _q q≥1, for classes of analytic functions studied by N. Ainulloev; these are generalizations (in a certain sense) of function sets introduced by L. V. Taikov. The exact values of their linear and Gelfandn-widths are obtained. The exact values of the Kolmogorov and Bernsteinn-widths of classes of analytic (in |z|<1) functions whose boundaryK-functionals are majorized by a prescribed functions are also obtained. Translated fromMatermaticheskie Zametki, Vol. 65, No. 2, pp. 186–193, February, 1999.     

On the existence of the best discrete approximation in norm by reciprocals of real polynomials

For the given data (w_i,x_i,y_i), i=1,…,M, we consider the problem of existence of the best discrete approximation in l_p norm (1≤p<∞) by reciprocals of real polynomials. For this problem, the existence of best approximations is not always guaranteed. In this paper, we give a condition on data which is necessary and sufficient for the existence of the best approximation in l_p norm. This condition is theoretical in nature. We apply it to obtain several other existence theorems very useful in practice. Some illustrative examples are also included.     

On the degree of approximation of functions belonging to a Lipschitz class by Hausdorff means of its fourier series

In a recent paper Lal and Yadov [4] obtained a theorem on the degree of approximation for a function belonging to the Lipschitz class Lipα using the product of the Cesàro and Euler means of order one of its Fourier series. In this paper we extend this result to any regular Hausdorff matrix for the same class of functions.     

Stability of the linear inequality method for rational Chebyshev approximation

The linear inequality method is an algorithm for discrete Chebyshev approximation by generalized rationals. Stability of the method with respect to uniform convergence is studied. Analytically, the method appears superior to all others in reliability.     

Distribution of poles of rational functions of best approximation

This article describes the construction of an entire function E(z) such that for any sequence of rational functions of best approximation to E(z) on the unit disc K, the corresponding set of poles { _nk ^* } is everywhere dense in the complement of K.Translated from Matematicheskie Zametki, Vol. 7, No. 3, pp. 289–293, March, 1970.     

Nonlinear approximation of functions based on nonnegative least squares solver

In computational practice, most attention is paid to rational approximations of functions and approximations by the sum of exponents. We consider a wide enough class of nonlinear approximations characterized by a set of two required parameters. The approximating function is linear in the first parameter; these parameters are assumed to be positive. The individual terms of the approximating function represent a fixed function that depends nonlinearly on the second parameter. A numerical approximation minimizes the residual functional by approximating function values at individual points. The second parameter's value is set on a more extensive set of points of the interval of permissible values. The proposed approach's key feature consists in determining the first parameter on each separate iteration of the classical nonnegative least squares method. The computational algorithm is used to rational approximate the function . The second example concerns the approximation of the stretching exponential function at $x\ge 0$ by the sum of exponents.     

Lp空间Kantorovich-Vertesi算子逼近的Jackson型估计

考虑了Kantorovich-Vertesi有理插值型算子L^*n,s（f,X,x）对L^p[-1,1]（1≤p≤∞）空间函数逼近的Jackson型估计。并获得了如下逼近阶：‖L^*n,s(f,X,x)-f(x)‖L^p[-1,1]≤Cp,sw(f,1/n 2)L^p[-1,1] （s＞2）。     

Legendre rational approximation on the whole line

The Legendre rational approximation is investigated. Some approximation results are established, which form the mathematical foundation of a new spectral method on the whole line. A model problem is considered. Numerical results show the efficiency of this new approach.     

OnM-ideals and best approximation

In this paper we will prove some theorems on theM-ideals of compact operators and the best approximation of quasitriangular operator algebras. These results improve and extend the known results in [4, 5, 7].This work is supported in part by the National Natural Science Foundation of China.