E.H Kaufman

Uniform Approximation by Rational Functions Having Restricted Denominators.

Abstract This paper considers approximation of continuous functions on a compact metric space by generalized rational functions for which the denominators have bounded coefficients and are bounded below by a fixed positive function. This lower bound alleviates numerical difficulties, and in some applications (e.g., digital filter design) has a useful physical interpretation. A “zero in the convex hull” characterization of best approximations is developed and used to prove uniqueness and de la Vallee Poussin results. Examples are given to illustrate this theory and its differences with the standard theory, where the denominators are merely required to be positive. A modified differential correction algorithm is presented and is proved to always converge at least linearly, and often quadratically.

A combined Remes-differential correction algorithm for rational approximation

Abstract : In this paper a hybrid Remes-differential correction algorithm for computing best uniform rational approximants on a compact subset of the real line is developed. This algorithm differs from the classical multiple exchange Remes algorithm in two crucial aspects. First of all, the solving of a nonlinear system to find a best approximation on a given reference set in each iteration of the Remes algorithm is replaced with the differential correction algorithm to compute the desired best approximation on the reference set. Secondly, the exchange procedure itself has been modified to eliminate the possibility of cycling that can occur in the usual exchange procedure. This second modification is necessary to guarantee the convergence of this algorithm on a finite set without the usual normal and sufficiently dense assumptions that exist in other studies. (Author)

Uniform approximation with rational functions having negative poles

Abstract A general theory of uniform approximation with rational functions having negative poles is developed. An existence theory is given and local characterization and uniqueness results are developed. Algorithms for computing these approximants are given, together with numerical results.

An adaptive differential-correction algorithm☆

A combined first Remes differential-correction algorithm for uniform generalized rational approximation with restricted range constraints is presented. This algorithm can be applied when the data sets are too large to allow for the direct use of differential correction and when the second Remes algorithm does not apply because of the lack of an alternating theory. Under the assumption that differential correction produces a good (though not necessarily best) approximation on each (small) subset to which it is applied, it is proven that the algorithm terminates in a finite number of steps at a good approximation on the entire data set. This is established even though, unlike the standard first Remes, the algorithm sometimes discards points in passing from one subset to the next. This theory also allows for the set to be infinite. Also, a discretization theorem is presented and the algorithm is illustrated with a numerical example.

Local and global Lipschitz constants

Abstract Let X be a closed subset of I = [−1, 1], and let B n ( f ) be the best uniform approximation to f ϵ C [ X ] from the set of polynomials of degree at most n . An extended global Lipschitz constant is defined for f, and it is shown that this constant is asymptotically equivalent to the strong unicity constant. Estimates of the size of the local Lipschitz constant for f are given when the cardinality of the set of extremal points of f − B n ( f )is n + 2. Examples which illustrate that the local and extended global Lipschitz constants may have very different asymptotic behavior are constructed.

Local Lipschitz constants

Abstract Let X be a closed subset of I= [− 1, 1], For f ϵ C[X], the local Lipschitz constant is defined to be λ nδ (f) = sup { ∥B n (f) − B n (g)∥ ∥f − g∥: 0 , where Bn(g) is the best approximation in the sup norm to g on X from the set of polynomials of degree at most n. It is shown that under certain assumptions the norm of the derivative of the best approximation operator at f is equal to the limit as δ → 0 of the local Lipschitz constant of f, and an explicit expression is given for this common value. The, possibly very different, characterizations of local and global Lipschitz constants are also considered.

A combined remes-differential correction algorithm for rational approximation: experimental results

Abstract It is our contention that the recently developed Remes-difcor algorithm is the best general purpose algorithm available for computing best uniform rational approximations, and should be widely used as a library routine. The purpose of this paper is to support this contention by theoretical arguments and by a numerical comparison of the Remes-difcor algorithm, the differential correction algorithm, and the widely-used Remes algorithm. The Remes-difcor algorithm is shown to be more robust than the Remes algorithm and faster than the differential correction algorithm. The three algorithms are briefly described and discussed, and the experimental results for 70 examples are presented in six tables.

Linear Convergence and the Bisection Algorithm

(1986). Linear Convergence and the Bisection Algorithm. The American Mathematical Monthly: Vol. 93, No. 1, pp. 48-51.

Uniform reciprocal approximation subject to linear constraints

Abstract In this paper a general theory is presented for uniform approximation by reciprocals of elements of a linear subspace subject to linear constraints. In fact, the main results in the known theory of constrained linear approximation have analogues in this non-linear setting.

Limit cycles for successive projections onto hyperplanes in Rn

Abstract In this paper we consider successive orthogonal projections onto m hyperplanes in R n, where m ⩾ 2 and n ⩾ 2. A limit cycle is defined to be a sequence of points formed by projecting onto each of the hyperplanes once in a prescribed order, with the last projection giving the starting point. Several examples, including triangles, quadrilaterals, regular polygons, and arbitrary collections of lines in R 2, are solved for the limit cycle. Limit cycles are found in various ways, including by a limiting process and by solving an mn × mn linear system of equations. The latter approach will produce all the limit cycles for an arbitrary ordered set of m hyperplanes in R n.