56013078
A theorem of Grace on the zeros of polynomials, revisited
117822564
Underpolynomials and infrapolynomials
121117321
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)
121796401
Equilibrium of inverse-distance forces in three-dimensions.
119998985
On the derivative of a polynomial and Chebyshev approximation
Introduction. The location of the zeros of the derivative of a polynomial has been much studied, as has the location of the zeros of the Chebys;hev polynomial. In ?1 of the present note we set forth in a direct and elementary manner the equivalence of these two problems in a suitably specialized situation. This conclusion is mentioned (for integral 'i) with an indication of the proof by Fekete and von Neumann [4],2 and the conclusion is related to a much deeper investigation due to Fekete [3]. We obtain (?9) some new results on zeros of approximating polynomials and (??2, 3, 8) on the argument of the deviation. In ?10 we consider approximation by an arbitrary linear family. Throughout the paper we study primarily approximation on a set of n points by a polynomial of degree n-2. In ??4-7 weight functions with infinities and Chebyshev rational functions are introduced.
42985124
BEST APPROXIMATORS WITHIN A LINEAR FAMILY ON AN INTERVAL.
It is our purpose in the present note to investigate the behavior of functions p(x) of best approximation to a given function f(x), where p(x) is supposed to belong to a linear family P and where deviation is measured on E:O O, w(x) = 1, and is now to be studied more generally (but not including the case p < 1). For positive r(t) we may write r (t) -= t(t), and our problem becomes that of variable-power approximation, an aspect that we do not stress for nonconstant a(t). We use r(t) directly to define our norm, in contrast to Orlicz who introduces a function comparable with our r(t) but defines quite differently a norm in the classical sense. We considered in reference' and consider particularly here properties of oscillation of f(x) - p(x) on E and identical vanishing of f(x) - p(x) on a subset of E of positive measure, extensions of classical results due to D. Jackson. Our previous results derived necessary conditions for best approximators, whereas here we treat also sufficient conditions. Under suitable conditions we prove the existence of a weight function so that a given p(x) shall be a best approximator. Our methods involve a certain algebraic inequality akin to orthogonality (inequality [1] below), of a type which was recently used by one of us2 in an invited address. The authors have previously investigated3 approximation on a finite set with the pth power norm, but postpone such consideration of the r-norm to another occasion. 1. We consider henceforth a deviation as defined by a generalized norm |g(x)|| =fEr(Ig(x) )w(x)dx, E:O < x < 1, where r(t) is defined for all t, 0 < t < o, where T"(t) exists and is continuous for such t, and where w(x) is continuous on E and positive a.e. We study for a given function
122697689
Finite-difference and finite-element methods of approximation
The finite-difference and finite-element methods for approximating the solution of elliptic boundary-value problems are discussed. The analysis of the order of accuracy is outlined, and the results compared, with some comment on special problems connected with singularities.
117015916
Numerical analysis : an introduction : based on a symposium organized by the Institute of Mathematics and Its Applications, Birmingham, England, 1965