G.D Taylor

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.

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.

Convex Lp approximation

In [7] Karlovitz developed an algorithm for finding best Lp approximations from certain finite-dimensional subspaces for p an even integer. It will be shown that this algorithm converges for 2 < p < co when the approximating subspace is replaced by certain closed convex subsets of a linitedimensional subspace. Furthermore, the restrictions which must be placed on the functions involved to ensure convergence are weakened. We shall consider the problem of approximating 0 by elements of a closed nonempty convex subset K which is contained in a finite-dimensional subspace and which does not contain 0. This is seen to be equivalent to the general problem of approximating a function f by elements of a closed convex subset G, with f 65 G and G contained in a finite-dimensional subspace, by simply translating all functions involved by -f. That is,

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.

Periodic quadratic spline interpolant of minimal norm

Abstract : It is shown that the periodic quadratic spline interpolant has minimal norm if the knots are equally spaced. (Author)

Chebyshev approximation by reciprocals of polynomials on [0, ∞)

Abstract : A uniform approximation theory is developed for the problem of approximating functions of the form f = B . g on the interval 0 < or = x < infinity with functions of the form B(x)/P(x), P a polynomial and B(x) fixed subject to certain requirements. In particular, an alernation theorem is developed which differs from the standard alternation theorem.

A unified theory of strong uniqueness in uniform approximation with constraints

Abstract The previously developed unified theory of constrained uniform approximation from a finite dimensional subspace is extended to treat strong uniqueness and continuity of the best approximation operator.

Approximation with reciprocals of polynomials on compact sets

Abstract : A theory best uniform approximation by reciprocals of polynomials on compact sets is developed. In particular an existence theorem is shown to hold in this case. (Author)

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.

Remarks on the Rank of Hermite–Birkhoff Interpolation

We pose the following extension of the real polynomial Hermite–Birkhoff interpolation problem : Given an incidence matrix, what is the rank of the associated system of point and derivative point evaluations? Bounds are obtained for both first order and general incidence matrices. Several examples provide motivation.