Geometry and Topology of Continuous Best and Near Best Approximations

The existence of a continuous best approximation or of near best approximations of a strictly convex space by a subset is shown to imply uniqueness of the best approximation under various assumptions on the approximating subset. For more general spaces, when continuous best or near best approximations exist, the set of best approximants to any given element is shown to satisfy connectivity and radius constraints.     

Best Chebyshev approximation from families of ordinary differential equations

Within the framework of the Meinardus and Schwedt theory ofnonlinear Chebyshev approximation we consider the approximationof functions and data by approximants generated from the solutionof parameter dependent initial value problems in ordinary differentialequations. New results are obtained for the uniqueness and characterizationof best approximations in terms of properties of the differentialequation. Some examples of new approximating families are givenalong with computed best approximations.     

SOME NONLINEAR APPROXIMATIONS FOR MATRIX-VALUED FUNCTIONS

Some nonlinear approximants, i.e., exponential-sum interpolation with equal distance or at origin, (0,1)-type, (0,2)-type and (1,2)-type fraction-sum approximations, for matrixvalued functions are introduced. All these approximation problems lead to a same form system of nonlinear equations. Solving methods for the nonlinear system are discussed.Conclusions on uniqueness and convergence of the approximants for certain class of functions are given.     

Existence and Uniqueness of Matrix Padé Approximants

For the problems of the left and right matrix Padé approximations, we give the necessary and sufficient conditions for the existence of their solutions. If the left Padé approximant exists, then we prove that its uniqueness is equivalent to the existence of right Padé approximants, and we further give the exact results about the dimension of the linear space $^LR^{*}(m,n)$ formed from the left Padé approximants.     

Optimality and uniqueness conditions in complex rational Chebyshev approximation with examples

It is well known that best complex rational Chebyshev approximants are not always unique and that, in general, they cannot be characterized by the necessary local Kolmogorov condition or by the sufficient global Kolmogorov condition. Recently, Ruttan (1985) proposed an interesting sufficient optimality criterion in terms of positive semidefiniteness of some Hermitian matrix. Moreover, he asserted that this condition is also necessary, and thus provides a characterization of best approximants, in a fundamental case.In this paper we complement Ruttan's sufficient optimality criterion by a uniqueness condition and we present a simple procedure for computing the set of best approximants in case of nonuniqueness. Then, by exhibiting an approximation problem on the unit disk, we point out that Ruttan's characterization in the fundamental case is not generally true. Finally, we produce several examples of best approximants on a real interval and on the unit circle which, among other things, give some answers to open questions raised in the literature.     

On the Problem of Best Rational Approximation with Interpolating Constraints (I)

The aim of this conjoint paper is to discuss the problem of the best rational approximation with interpolating constraints. In part I, we give the necessary and sufficient conditions for the existence of the best rational approximations and establish some characterization theorem for such approximations. The problems of uniqueness, the properties for the set of best approximations, strong uniqueness and continuity of best approximation operator are considered in Part II. The results obtained in this paper are the completion and extension of those given in [1].     

An algorithm for approximation by elements of a cone in a banach space

Given a closed convex cone of approximants in a strictly convex reflexive Banach space we give an implementable, iterative algorithm for finding the best approximation for a point outside this cone. First the necessary duality theorems are proved and then the characterizations of the best approximations are given. Convergence of the algorithm is then proved. Some numerical results are also included.     

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 Usual Behavior of Rational Approximation, II

The speeds of convergence of best rational approximations, best polynomial approximations, and the modulus of continuity on the unit disc are compared. We show that, in a Baire category sense, it is expected that subsequences of these approximants will converge at the same rate. Similar problems on the interval [−1, 1] are also examined. A problem raised by P. Turán (J. Approx. Theory29, 1980, 23-89) concerning rational approximation to non-analytically continuable ƒ on the unit circle is negated as an application.     

Unicity of best one-sided L1-approximations for certain classes of spline functions

The chief purpose of this paper is to present conditions ensuring uniqueness of best one-sided L¹-approximations for approximation by finite dimensional subspaces of differentiable functions. Using these results it is shown that uniqueness of such best approximations will hold for a class of generalized spline subspaces and also for spline subspaces satisfying certain boundary conditions. These considerations have an important application to uniqueness of quadrature formulae of “highest possible degree of precision”.     

Best Uniform Approximation of Complex-Valued Functions by Generalized Polynomials Having Restricted Ranges

We study the uniform best restricted ranges approximations of complex-valued functions by generalized polynomials. The theory, generalizing the real-valued case, embraces the theorems of existence, characterization, uniqueness, and strong uniqueness.     

Some aspects of two-point Padé approximants

Recently McCabe and Murphy have considered the two-point Padé approximants to a function for which (formal) power series expansions at the origin and at infinity are given. In this paper these approximations are slightly modified and determinant representations for them are given. The existence of various three-term recursion relations for the numerators and denominators of these approximants is shown. Based on these, a new continued fraction representation for these approximants is obtained and also an efficient recursive method is proposed for the determination of the coefficients of all the approximants that obtain from a given number of terms of the power series.     

Minimum relative error approximations for 1/t

Summary We give explicit solutions to the problem of minimizing the relative error for polynomial approximations to 1/t on arbitrary finite subintervals of (0, ). We give a simple algorithm, using synthetic division, for computing practical representations of the best approximating polynomials. The resulting polynomials also minimize the absolute error in a related functional equation. We show that, for any continuous function with no zeros on the interval of interest, the geometric convergence rates for best absolute error and best relative error approximants must be equal. The approximation polynomials for 1/t are useful for finding suitably precise initial approximations in iterative methods for computing reciprocals on computers.     

基于广义逆的矩阵Padé逼近的De Montessus-De Ballore型收敛性定理

基于广义逆的矩阵Padé 逼近^[4,5]的一个行收敛性定理,即著名的De Montessus-De Ballore回收敛定理在本文首次得以建立,根据这一结果,唯一性定理被简洁地证明,并获得一个实用的存在性定理,     

Positive weak solutions of semilinear second order elliptic inequalities via variational inequalities

Existence, uniqueness and convergence of approximants of positive weak solutions for semilinear second order elliptic inequalities are obtained. The nonlinearities involved in these inequalities satisfy suitable upper or lower bound conditions or monotonicity conditions. The lower bound conditions are allowed to contain the critical Sobolev exponents. The methodology is to establish variational inequality principles for demicontinuous pseudo-contractive maps in Hilbert spaces by considering convergence of approximants and apply them to the corresponding variational inequalities arising from the semilinear second order elliptic inequalities. Examples on the existence, uniqueness and convergence of approximants of positive weak solutions of the semilinear second order elliptic inequalities are given.     

Polynomials of least deviation from zero

The problem of uniqueness of best approximations is analyzed in the multidimensional case. In the space C, a set of polynomials of least deviation from zero is constructed.     

Best Approximation and Cyclic Variation Diminishing Kernels

We study best uniform approximation of periodic functions from

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where the kernelK(x, y) is strictly cyclic variation diminishing, and related problems including periodic generalized perfect splines. For various approximation problems of this type, we show the uniqueness of the best approximation and characterize the best approximation by extremal properties of the error function. The results are proved by using a characterization of best approximants from quasi-Chebyshev spaces and certain perturbation results.     

Uniform reciprocal approximation subject to coefficient constraints

In this paper best approximation by reciprocals of functions of a subspace U_n=span (u₁,...,u_n) satisfying coefficient constraints is considered. We present a characterization of best approximations. When (u₁,...,u_n) is a Descartes system an explicit characterization of best approximations by equioscillations is given. Existence and uniqueness results are shown. Moreover, the theory is applied to best approximaitons by reciprocals of polynomials.     

A note on strong uniqueness constants

We expand a result of Blatt, concerning the strong uniqueness constants of uniform best approximations on [− 1, 1].     

Generalized matrix Pade approximants

For rectangular matrix functions with restricted sizes of their column and row, we introduce the problem of Padé approximation similar to its scalar counterpart. Results on the existence and uniqueness of the approximants are given. Determinantal expressions and some properties of the approximants are established. Supported by Science Fund for Youth of Chinese Academy of Sciences.