Nonuniqueness of best rational Chebyshev approximations on the unit disk

AbstractIn the past it has been unknown whether complex rational best Chebyshev approximations (BAs) on the unit disk need be unique. This paper answers this and related questions by exhibiting examples in which: (a) the BA is not unique, (b) the number of distinct BAs is arbitrarily large, (c) the BA to a real analytic function f (i.e., f(z) = f(z)) among rational functions with real coefficients is not unique, and (d) the complex BAs to such a function are better than any approximation with real coefficients. Except in case (d), our constructions hold for approximation of arbitrary type (m, n) with n ⩾ 1. Finally, by the same methods we also establish the new result that if a function is approximated on a small disk about 0 of radius ε (or on an interval of length ε), then as ε → 0, the BA need not in general approach the corresponding Padé approximant in a sense considered by J. L. Walsh

Approximation with sums of exponentials in Lp[0, ∞)

AbstractWe consider the problem of approximating a given f from Lp [0, ∞) by means of the family Vn(S) of exponential sums; Vn(S) denotes the set of all possible solutions of all possible nth order linear homogeneous differential equations with constant coefficients for which the roots of the corresponding characteristic polynomials all lie in the set S. We establish the existence of best approximations, show that the distance from a given f to Vn(S) decreases to zero as n becomes infinite, and characterize such best approximations with a first-order necessary condition. In so doing we extend previously known results that apply in Lp[0, 1]

A differential correction algorithm for exponential curve fitting / CAC No. 92

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