Best Approximation Of Functions Research Articles

Best approximation of functions specified with an error at a finite number of points

It is proved that for convex and centrally symmetric classes of functions a linear method is included among the best (in a definite sense) methods of approximation from values specified with an error at a finite number of points. For some of the simplest classes linear best methods are constructed and their error is estimated.

Features of the best approximation of functions by polynomials on small computers

Features of the best approximation of functions by polynomials on small computers

Padé approximants as limits of rational functions of best approximation, real domain

Padé approximants as limits of rational functions of best approximation, real domain

The numerical range of functions and best approximation

In this note, we state general conditions which imply that the numerical range of a function mapping a setSinto a normed linear spaceYis the closed convex hull of the spatial numerical range of the function. This conclusion is of interest since, for example, it is equivalent to an extension to non-compact spaces of Kolmogoroff's characterization of functions of best approximation.

On estimating best approximations of functions defined by integral equations

On estimating best approximations of functions defined by integral equations

On the convergence of rational functions of best approximation to a meromorphic function

On the convergence of rational functions of best approximation to a meromorphic function

Note on the Convergence of Sequences of Rational Functions of Best Approximation to a Meromorphic Function

If the function f(z) is meromorphic at every finite point of the plane, analytic on C: [unk]z[unk] </= 1, there are results that can be established concerning the convergence of the sequence R(n)(z) of rational functions of each degree n of best approximation to f(z) on C. However, we need to assume certain additional conditions on the convergence of the sequence R(n)(z) itself. The results generalize to more general sets C and to more general functions.

The Convergence of Rational Functions of Best Approximation to the Exponential Function. II

Let ${W_{m,n}}(z)$ be a rational function of type (m, n) of best uniform approximation to the function ${e^z}$ on the closed unit disk. In this paper we show that any sequence $\{ {W_{m,n}}(z)\}$ for which $m + n \to \infty$ must converge to ${e^z}$ for all values of z. This is the first result which describes completely the regions of convergence of arbitrary sequences formed from a Walsh array.

BEST APPROXIMATIONS OF FUNCTIONS IN THELpMETRIC BY HAAR AND WALSH POLYNOMIALS

In this work the modulus of continuity of functions in the Lp metric (1 ≤ p < ∞) is estimated through its best approximations in this metric by Haar and Walsh polynomials. Besides, estimates of best approximations of functions by Haar and Walsh polynomials in the Lq metric are obtained by the same approximations in the Lp metric (1 ≤ p ≤ ∞). In the last case, the results are analogous to those which were proved for approximations by trigonometric polynomials by P. L. Ul'janov and also by S. B. Steckin and A. A. Konjuskov. Bibliography: 26 items.

The convergence of rational functions of best approximation to the exponential function. II

Let ${W_{m,n}}(z)$ be a rational function of type (

Some Examples in Degree of Approximation by Rational Functions

We exhibit examples of (1) series that converge more rapidly than any geometric series where the function represented has a natural boundary, (2) the convergence of a series with maximum geometric degree of convergence yet having limit points of poles of the series everywhere dense on a circumference in the complement of $E$, (3) a Padé table for an entire function whose diagonal has poles every-where dense in the plane and (4) a corresponding example for the table of rational functions of best approximation of prescribed type.

Some examples in degree of approximation by rational functions

We exhibit examples of (1) series that converge more rapidly than any geometric series where the function represented has a natural boundary, (2) the convergence of a series with maximum geometric degree of convergence yet having limit points of poles of the series everywhere dense on a circumference in the complement of E E , (3) a Padé table for an entire function whose diagonal has poles every-where dense in the plane and (4) a corresponding example for the table of rational functions of best approximation of prescribed type.

The Convergence of Rational Functions of Best Approximation to the Exponential Function

The Convergence of Rational Functions of Best Approximation to the Exponential Function

The convergence of rational functions of best approximation to the exponential function

The object of the paper is to establish convergence throughout the entire complex plane of sequences of rational functions of prescribed types which satisfy a certain degree of approximation to the function a e y z a{e^{yz}} on the disk | z | ≦ ρ |z| \leqq \rho . It is assumed that the approximating rational functions have a bounded number of free poles. Estimates are given for the degree of best approximation to the exponential function by rational functions of prescribed types. The results obtained in the paper imply that the successive rows of the Walsh array for a e y z a{e^{yz}} on | z | ≦ ρ |z| \leqq \rho converge uniformly to a e y z a{e^{yz}} on each bounded subset of the plane.

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 $$\{ \mathop r\limits^* _n (z)\} $$ of rational functions of best approximation to E(z) on the unit disc K, the corresponding set of poles {α nk * } is everywhere dense in the complement of K.

On estimating best approximations of functions defined by differential equations

Finite difference methods are used to estimate the error in approximations to functions defined by differential equations. The problem of minimising the maximum of this error estimate can be solved by linear programming in the linear case, and by a method due to the authors in the nonlinear case. It is shown by examples that this new approach can improve substantially on techniques which minimise the maximum residual in the differential equation, and a convergence result as the mesh spacing tends to zero is given for the linear case.

Best approximations of functions of several variables by sums of functions of fewer variables

The minimum of the mean-square deviation of approximations of functions of several independent variables by sums of functions of fewer variables in a multidimensional parallelepiped is investigated. The approximating function yielding the minimum mean-square deviation is obtained and this minimum deviation is calculated.

On the application of Chebyshev's methods to the problem of balancing mechanisms

The aim of this book is to present the theory of the best approximation of functions in an engineering context. While the detailed presentation and application of the best uniform approximation theory of P.L. Chebyshev is its primary direction, considerable effort is made to present the best average approximation theory for comparison. Application of these theories is centered about the balancing of mechanisms. The excerpts presented here start with the mathematical criteria of Chebyshev's theory. They continue with discussions of the best average and best uniform approximations to a unit circle by means of a constant. Furthermore, iterative and graphical approximation methods for the determination of the first order harmonic which deviates least from a given function in the sense of Chebyshev are shown. Finally, each of the theoretical results is applied to a specific problem of mechanism balancing.

On best approximations of functions of two variables

On best approximations of functions of two variables

Mean approximation of functions by Fourier-Gegenbauer sums

Necessary and sufficient conditions for best approximations of functions in the\(L_{(1 - X^2 )^\alpha }^2 \) (−1,1) metric, −1/2≤α<1/2 to zero at a certain rate are established (for α=−1/2 known results are obtained). Inequalities for algebraic polynomials are used in the reasoning.