Method of Generalized Moment Representations in the Theory of Rational Approximation (A Survey)

We give a survey of the method of generalized moment representations introduced by Dzyadyk in 1981 and its applications to Padé approximations. In particular, some properties of biorthogonal polynomials are investigated and numerous important examples are given. We also consider applications of this method to joint Padé approximations, Padé–Chebyshev approximations, Hermite–Padé approximations, and two-point Padé approximations.     

Uniformn-analytic polynomial approximations of functions on rectifiable contours in ℂ

We study approximations of functions byn-analytic polynomials in the uniform norm on closed rectifiable Jordan curves in the complex plane. It is shown that, in contrast to the case of uniform approximations by complex polynomials, there are no topological criteria for the existence of such approximations. We obtain a criterion for the existence ofn-analytic polynomial approximations in terms of analytic properties of these curves. Translated fromMatematicheskie Zametki, Vol. 59, No. 4, pp. 603–609, April, 1996. The author is extremely grateful to A. G. Vitushkin and P. V. Paramonov for the statement of the problem and their attention to the work. The work was partially supported by the Russian Foundation for Basic Research under grant No. 93-01-00225.     

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.     

Approximation of classes of periodic functions with small smoothness

We prove that the approximations of classes of periodic functions with small smoothness in the metrics of the spaces C and L by different linear summation methods for Fourier series are asymptotically equal to the least upper bounds of the best approximations of these classes by trigonometric polynomials of degree not higher than (n - 1). We establish that the Fejér method is asymptotically the best among all positive linear approximation methods for these classes.     

Projection methods and approximations for ordinary differential equations

A formulation of a differential equation as projection and fixed point problem allows approximations using general piecewise functions. We prove existence and uniqueness of the approximate solution, convergence in the L₂ norm and nodal superconvergence. These results generalize those obtained earlier by Hulme for continuous piecewise polynomials and by Del four-Dubeau for discontinuous piecewise polynomials. A duality relationship for the two types of approximations is also given. This research has been supported in part by the Natural Sciences and Engineering Research Council of Canada (Grant OGPLN-336) and by the “Ministère de l’Education du Québec” (FCAR Grant-ER-0725).     

Approximation of Infinitely Differentiable Periodic Functions by Interpolation Trigonometric Polynomials in Integral Metric

We obtain asymptotic equalities for the upper bounds of approximations of periodic infinitely differentiable functions by interpolation trigonometric polynomials in the metric of L ₁ on the classes of convolutions.     

Approximation of Certain Classes of Singular Integrals by Algebraic Polynomials

We study the problem of pointwise approximation by algebraic polynomials for classes of functions that are singular integrals of bounded functions. We obtain asymptotically exact estimates of approximations.     

Asymptotic Approximations to the Nodes and Weights of Gauss–Hermite and Gauss–Laguerre Quadratures

Asymptotic approximations to the zeros of Hermite and Laguerre polynomials are given, together with methods for obtaining the coefficients in the expansions. These approximations can be used as a stand‐alone method of computation of Gaussian quadratures for high enough degrees, with Gaussian weights computed from asymptotic approximations for the orthogonal polynomials. We provide numerical evidence showing that for degrees greater than 100, the asymptotic methods are enough for a double precision accuracy computation (15–16 digits) of the nodes and weights of the Gauss–Hermite and Gauss–Laguerre quadratures.     

Approximation of Periodic Analytic Functions by Interpolation Trigonometric Polynomials in the Metric of the Space L

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

Approximation by Interpolation Trigonometric Polynomials in Metrics of the Space Lp on the Classes of Periodic Entire Functions*

Ukrainian Mathematical Journal - We establish asymptotic equalities for the least upper bounds of approximations by interpolation trigonometric polynomials with equidistant distribution of...     

Approximation of the Lerch Zeta-Function

We consider uniform in parameters approximations of the Lerch zeta-function by Dirichlet polynomials. It allows us to obtain uniform in parameters bounds in the critical strip.     

Approximation properties of the Vallée-Poussin means of partial sums of a mixed series of Legendre polynomials

We estimate the order of weighted approximations of functions and their derivatives by using the means of mixed series of Legendre polynomials. As the main result, we obtain estimates of the order of approximation of a function and its derivatives by the Vallé-Poussin means and their derivatives.     

Asymptotic analysis of the Askey-scheme I: from Krawtchouk to Charlier

We analyze the Charlier polynomials C _n (χ) and their zeros asymptotically as n → ∞. We obtain asymptotic approximations, using the limit relation between the Krawtchouk and Charlier polynomials, involving some special functions. We give numerical examples showing the accuracy of our formulas.      

Approximation by Classical Orthogonal Polynomials with Weight in Spaces L2,γ(a,b) and Widths of Some Functional Classes

Russian Mathematics - We investigate approximations of functions of classes W2(Dγ;(a,b)), r = 2, 3, …, by classical orthogonal polynomials with a weight γ in the spaces...     

Adaptive wavelet methods for elliptic partial differential equations with random operators

We apply adaptive wavelet methods to boundary value problems with random coefficients, discretized by wavelets in the spatial domain and tensorized polynomials in the parameter domain. Greedy algorithms control the approximate application of the fully discretized random operator, and the construction of sparse approximations to this operator. We suggest a power iteration for estimating errors induced by sparse approximations of linear operators.     

A Poisson–Charlier approximation for nonstationary queues

In this paper, we develop a new approximation for nonstationary multiserver queues with abandonment. Our method uses the Poisson–Charlier polynomials, which are a discrete orthogonal polynomial sequence that is orthogonal with respect to the Poisson distribution. We show that by appealing to the Poisson–Charlier polynomials that we can estimate the mean, variance, and probability of delay of our nonstationary queueing system with good accuracy. Lastly, we provide a numerical example that illustrates that our approximations are effective.     

Equidistribution of points via energy

We study the asymptotic equidistribution of points with discrete energy close to Robin’s constant of a compact set in the plane. Our main tools are the energy estimates from potential theory. We also consider the quantitative aspects of this equidistribution. Applications include estimates of growth for the Fekete and Leja polynomials associated with large classes of compact sets, convergence rates of the discrete energy approximations to Robin’s constant, and problems on the means of zeros of polynomials with integer coefficients.     

On a convergence of the rational-trigonometric-polynomial approximations realized by the roots of the Laguerre polynomials

The paper considers a problem of approximation of functions by means of their finite number of Fourier coefficients. Convergence acceleration of approximations by the truncated Fourier series is achieved by application of polynomial and rational correction functions. Rational corrections include unknown parameters whose determination is a crucial problem. We consider an approach connected with the roots of the Laguerre polynomials and study the rates of convergence of such approximations.     

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.     

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.