Tchebycheff Approximation Research Articles

On the distance between polytopes

This paper is written to fill an apparent gap in the literature in connection with some elementary problems of distance in finite dimensional normed linear spaces. In particular the problem of determining the distance between two polytopes in a normed n n -dimensional real vector space is considered. Special consideration is given to the case in which the norm is a twice differentiable function of its arguments and for this case a convex programming algorithm is presented. In addition several other cases are considered including the well known discrete Tchebycheff approximation. The results should find application in approximation theory.

Applications of the Hahn-Banach Theorem in Approximation Theory

Applications of the Hahn-Banach Theorem in Approximation Theory

Approximation from a curve of functions

for all ?-polynomials P,(A, T) of order n. Here I[ lip denotes the norm in Lp [0, 1]. A solution is a best Lp approximation to f . Not unexpectedly, this problem does not always have a solution. The concept of ?-polynomial is extended in section 2 and three existence theorems (Theorems 3, 4 and 5) are established. The first, Theorem 3, is for 1 <p__< 0% I compact and ?(t, x) sufficiently differentiable. The important case when I is not compact is considered in Theorem 4 for 1 <p < oo and Theorem 5 for p = oo. In section 3 we note the interesting fact that if F spans Lp [0, 1] (as in the above examples), then the error of the best Lp approximation, for l < p < o o , strictly decreases as a function of n until it reaches zero. The final section considers Tchebycheff approximation when P,(A, T, x) is analytic in t and a continuous complex valued function of the real variable x. Under the hypothesis that ?(t, x) is strictly totally positive (see [2], e tx, x t qualify)

Convergence of Best Rational Tchebycheff Approximations

Convergence of Best Rational Tchebycheff Approximations

Existence of best rational Tchebycheff approximations

Existence of best rational Tchebycheff approximations

Convergence of best rational Tchebycheff approximations

Cf. Achieser [1]. In this paper, we investigate the behavior of the error functional E(n, m, f), in terms of the continuity properties of the function f, for various hypotheses on the growth of the degrees n and m of the numerator and denominator polynomials. ?2 generalizes some results of Walsh [2] concerning the conditions under which E(n, m, f) approaches zero with increasing n and m, while ?3 gives conditions uhder which E(n, m, f) does not approach zero; the results in these sections are shown to have counterparts when other fundamental Markoff systems in C[a,b] replace the system { 1, x, x2, * * * }. A theorem of Jackson [3] on Tchebycheff approximation by polynomials states that

On the existence and characterization of best nonlinear Tchebycheff approximations

On the existence and characterization of best nonlinear Tchebycheff approximations

On the Existence and Characterization of Best Nonlinear Tchebycheff Approximations

On the Existence and Characterization of Best Nonlinear Tchebycheff Approximations

Note on Convex Programming

Introduction. In their paper, Newton's Method for Convex Programming and Tchebycheff Approximation [1], E. WV. Cheney and A. A. Goldstein consider the following problem: Given a convex continuous function F defined on a closed convex subset K of En, obtain (if such exists) a point x of K such that F(x) ? F(y) for all y in K. Replacing the convex hyperplanes by their support planes, one gets the general problem of convex programming. Let Q and A be two compact sets of En, b and c be two continuous real valued functions defined respectively on Q and A such that

Tchebycheff approximation in several variables

1. Introduction. The theory of Tchebycheff approximation for functions of one real variable has been understood for some time and is quite elegant. For about fifty years attempts have been made to generalize this theory to functions of several variables. These attempts have failed because of the lack of uniqueness of best approximations to functions of more than one variable. Indeed, Mairhuber [6] has shown that best approximations cannot be unique unless our functions are defined on a space homeomorphic to a subset of the unit circle. Further negative results have been obtained by Rivlin and Shapiro [11] and the conclusion is that there is no interesting Tchebycheff approximation problem in several variables for which best approximations are unique. Schoenberg [12] gives an account of a theory in several variables, but his assumptions essentially limit the applicability to functions of one variable. In this paper two theories of Tchebycheff approximation to functions of several variables are developed. It cannot be said that they are as satisfying as the theory for one variable, but they are workable theories. For the first theory the concept of a critical point set is given (the definition is somewhat different from that in one variable) and it is shown that the set of critical point sets of a best approximation is unique. A characterization theorem for best approximations is given in terms of critical point sets and other theorems normally found in the theory of Tchebycheff approximation are valid. Similar viewpoints have recently been investigated by Lawson [16] and Rivlin and Shapiro [17]. The second theory defines the strict approximation for functions defined on a finite point set. In addition to the results obtained from the first theory, the strict approximation is unique. It is also a best approximation in the usual sense. There are two basic features of the one variable theory which do not generalize readily. The first of these is the idea of a Tchebycheff set. No counterpart of these sets exists in this paper. A heuristic argument is given that shows that there is no possibility of defining Tchebycheff sets in several variables with the important properties present in the case of one variable. The second feature which does

Zeros of the Error Function for Tchebycheff Approximation in a Small Region†

Abstract : It is shown that for a given analytic funtion and a suitably restricted point set in a neighborhood of a given point there are at least n + 1 zeros of the error function near the point set. (Author)

Tchebycheff approximation in locally convex spaces

Tchebycheff approximation in locally convex spaces

Tchebycheff Approximations by Functions Unisolvent of Variable Degree

Tchebycheff Approximations by Functions Unisolvent of Variable Degree

Tchebycheff approximations by functions unisolvent of variable degree

1. The first steps toward a theory of nonlinear Tchebycheff approximations were made by T. S. Motzkin [2] and L. Tornheim [4]. They introduced properties of unisolvence and equivalent concept of n-parameter families. For approximating functions which have these properties an elegant theory of Tchebycheff approximations may be developed. Unfortunately, this class of functions does not contain any well-known functions, except for degenerate case of linear approximating functions and transformations thereof. Motzkin [3] has shown, however, that essentially nonlinear unisolvent functions do exist. In this paper a property weaker than unisolvence will be introduced which also allows development of an elegant theory of Tchebycheff approximations. The class of functions possessing this property contains many elementary nonlinear approximating functions. The main definitions are given in ?2. Existence and uniqueness are discussed in ?3 and a theorem on characterization of best Tchebycheff approximations is given. Also given is an interesting theorem on a topological property of parameter space. The final section contains some examples which illustrate ideas of paper. 2. Euclidean n-dimensional space is denoted by E.; points in E. are denoted by a, b, etc. and coordinates of a are (a', a2, * * , an). Curly brackets, { I, denote a set and {xl .. *} is read as the set of x such that .. All maxima and minima are taken over xe [0, 1] unless otherwise stated. The real function F= F(a, x) is defined for xE [0, 1] and aGP where P is a nonvoid subset of En. F is continuous in sense that given aoCP, xoG[0, 1] and e>O there is a 8>0 such that aEP, xe[0, 1], lao-al + xo x <5 implies that I F(ao, xo) F(a, x) I <e. f(x) will denote a continuous function on [0, 1]. The Tchebycheff Approximation Problem for f(x) may be stated as: Determine a*EP such that max |F(a*, x) -f(x) I _ max |F(a, x) -f(x) for all aSP. A solution, F(a*, x), to approximation problem is called a best approximation to f(x).

Note on Jordan elimination, linear programming and Tchebycheff approximation

Note on Jordan elimination, linear programming and Tchebycheff approximation

Sequence transformations based on Tchebycheff approximations

Sequence transformations based on Tchebycheff approximations

The Characterization of Best Nonlinear Tchebycheff Approximations

The Characterization of Best Nonlinear Tchebycheff Approximations

The characterization of best nonlinear Tchebycheff approximations

1.1. Let two real functions f and F be given such that f=f(x) is a continuous function for x [0, 1] and F== F(al, a2, * * *, a n, x) is a continuous function of n parameters and x [0, 1]. For simplicity of notation the point (a', a2, * * , an) in Euclidean n-space, En, is denoted by a. The domain of the parameters of F is denoted by P, a subset of En. The main problem in the theory of approximation of continuous functions may be stated as follows: Determine a*CP so that the deviation of the function F(a, x) from f(x) shall be minimized. Naturally one must define the deviation of F(a, x) from f(x) and different definitions lead to different theories. In this paper the deviation of F(a, x) from f(x) is taken to be maxxE[o,l] I F(a, x) -f(x) I. All maxima and minima are taken over xE [0, 1 ] unless otherwise stated. F(a*, x) is said to be a best approximation to f(x) if max I F(a*, x) -f(x) I _ max | F(a, x) -f(x) I for all aCEP. The results of this

Newton's method for convex programming and Tchebycheff approximation

w 1. Introduction. The rationale of Newton's method is exploited here in order to develop effective algorithms for solving the following general problem: given a convex continuous function F defined on a closed convex subset K of E,~, obtain (if such exists) a point x of K such that F(x)_<_F(y) for all y in K. The manifestation of Newton's method occurs when, in the course of computation, convex hypersurIaces are replaced by their support planes. The problems of infinite systems of linear inequalities and of infinite linear programming are subsumed by the above problem, as are certain Tchebycheff approximation problems for continuous functions on a metric compactum. In regard to the latter, special attention is devoted in w167 27--30 to the feasibility of replacing a continuum by a finite subset in such a way that a discrete approximation becomes an accurate substitute for the continuous approximation.

On the Convergence of an Algorithm for Best Tchebycheff Approximations

Next article On the Convergence of an Algorithm for Best Tchebycheff ApproximationsJ. R. RiceJ. R. Ricehttps://doi.org/10.1137/0107011PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] N. I. Achieser, Theory of approximation, Translated by Charles J. Hyman, Frederick Ungar Publishing Co., New York, 1956x+307 MR0095369 0072.28403 Google Scholar[2] C. de la Vallée Poussin, Leçons sur l'Approximation des Fonctions d'une Variable Réelle, Gauthier-Villars, Paris, 1919 Google Scholar[3] C. de la Vallée Poussin, Sur la méthode de l'approximation minimum, Ann. Soc. Sci. Bruxelles, Ser. I, 35 (1911), 1–16 Google Scholar[4] A. A. Goldstein and , Ward Cheney, A finite algorithm for the solution of consistent linear equations and inequalities and for the Tchebycheff approximation of inconsistent linear equations, Mathematical pre-print series No. 7, Convair Astronautics, San Diego, 1957 0084.01902 Google Scholar[5] T. S. Motzkin and , J. L. Walsh, Least pth power polynomials on a finite point set, Trans. Amer. Math. Soc., 83 (1956), 371–396 MR0081991 0072.25206 Google Scholar[6] Abe Shenitzer, Chebyshev approximation of a continuous function by a class of functions, J. Assoc. Comput. Mach., 4 (1957), 30–35 MR0093892 CrossrefISIGoogle Scholar[7] S. I. Zuhovickii˘, An algorithm for the solution of the Čebyšev approximation problem in the case of a finite system of incompatible linear equations, Doklady Akad. Nauk SSSR (N.S.), 79 (1951), 561–564, See also this Journal, 6 (1958), pp. 233–239 MR0043559 Google Scholar Next article FiguresRelatedReferencesCited byDetails On $L_1 $ Approximation I: Computation for Continuous Functions and Continuous Dependence14 July 2006 | SIAM Journal on Numerical Analysis, Vol. 4, No. 1AbstractPDF (1337 KB)Methods—Old and New—for Solving the Tchebycheff Approximation ProblemE. Stiefel14 July 2006 | Journal of the Society for Industrial and Applied Mathematics Series B Numerical Analysis, Vol. 1, No. 1AbstractPDF (1029 KB)Overdetermined Systems of Linear Equations18 July 2006 | SIAM Review, Vol. 5, No. 1AbstractPDF (1162 KB) Volume 7, Issue 2| 1959Journal of the Society for Industrial and Applied Mathematics History Submitted:10 September 1958Published online:10 July 2006 InformationCopyright © 1959 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0107011Article page range:pp. 133-142ISSN (print):0368-4245ISSN (online):2168-3484Publisher:Society for Industrial and Applied Mathematics