Charles B. Dunham

Best mean rational approximation

SummaryThe properties of best rational approximations with respect to a generalized integral norm on [a, b], which includes allLpnorms, 1≤p<∞, are considered. A necessary condition for an approximation to be (locally) best is obtained. A lower bound is given on the number of sign changes of the error curve or the number of points of interpolation. The problem of when a best approximation by polynomial rational functions can be degenerate is studied: it is shown that iff is analytic, degenerater is best tof only iff=r. Some non-uniqueness results for approximation of odd and even functions by polynomial rational functions are given. An example is given in which a local minimum of the error is not a global minimum.ZusammenfassungBetrachtet werden die Eigenschaften der besten rationalen Approximation mit Rücksicht auf ein verallgemeinertes Integralnorm auf [a, b], das einLp-Norm einschließt 1≤p<∞, erhalten wird eine notwendige Bedingung, wenn eine Approximation (lokal) am besten ist. Gegeben wird eine untere Grenze für die Anzahl der Zeichenwechsel der Fehlerkurve oder für die Anzahl der Interpolationspunkte. Untersucht wird das Problem, wann eine beste Approximation bei rationalen Polynomfunktionen entartet sein kann: es wird gezeigt, daß, wennf analytisch ist, ein entartetesr am besten zuf nur dann ist, wennf=r. Einige nicht eindeutige Resultate für Approximationen von geraden und ungeraden Funktionen bei rationalen Polynomfunktionen werden gegeben. Ein Beispiel ist angeführt, in dem ein lokales Fehlerminimum nicht ein globales Minimum ist.

Chebyshev Approximation with the Local Haar Condition

Chebyshev approximation by nonlinear families on a general compact space is studied. Attention is restricted to approximants satisfying a local Haar condition. A necessary and sufficient condition for the approximant to be locally best is given. A linear approximation problem is given which is equivalent to the nonlinear problem of locally best approximation. The existence of a minimal set on which a locally best approximation is locally best is shown. An alternation result is given for approximation on an interval.

Chebyshev approximation by rationals with constrained denominators

Abstract Approximating families of rational functions can be made nicer (tamed) by constraining the denominators below and above. Topological properties are improved, but characterization and uniqueness are more difficult for non-interior points.

Rate of convergence of discretization in Chebyshev approximation

The paper treats, in a particularly simple fashion, the practical problem of the rate of convergence of discretization in real and complex Chebyshev approximation. Both linear and nonlinear approximations are discussed and, subject to certain conditions, quadratic convergence of the discretizations is obtained along with an explicit rate constant which can be estimated numerically.

Approximation with one (or few) parameters nonlinear

Abstract In the case of one (or few) nonlinear parameters, minimax approximation can be done by applying minimization with respect to the nonlinear parameters and reduction to a linear subproblem. Only the case of a linear subproblem with a Haar subspace is considered, in which case a linear Remez algorithm is used.

Minimax Approximation by a Semi-Circle

A continuous function on a ray

Convergence of the Fraser-Hart algorithm for rational Chebyshev approximation

[0,\alpha ]

Feasibility of “perfect” function evaluation

is approximated by a convex semi-circle. This is related to the fabrication of an aspheric lens surface.

The limit of non-linear Chebyshev approximation on subsets

The Fraser-Hart variant of the Remez algorithm is used to determine the best rational Chebyshev approximation to a continuous function on an interval. A necessary and suffilcient condition for the matrix of the associated linear system to be nonsingular at.the solution to the approximation problem is given. It is shown that the Fraser-Hart method may fail even if started arbitrarily close to the solution of the approximation problem. Use of the secant method in place of the Fraser-Hart iteration is also considered.

Approximation by alternating families on subsets

This paper casts doubt on the feasibility of perfect subroutines or perfect hardware for mathematical functions, in particular by conversion from double precision. It suggests a slightly relaxed standard.