TL;DR
This paper revisits classical rational approximation problems related to the Sign function, extending known solutions to more complex cases and analyzing their properties.
Contribution
It uncovers overlooked solutions in the approximation of the Sign function over multiple intervals and studies their mathematical properties.
Findings
Explicit solutions for three-band approximation problems
Identification of previously overlooked solutions
Analysis of properties of these solutions
Abstract
The best uniform rational approximation of the Sign function on two intervals separated by zero was explicitly found by E.I. Zolotar\"ev in 1877. The natural extension of this problem to three bands was solved by E.Stiefel in 1961. We indicate the solutions overlooked by the prominent geometer and study their properties.
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Taxonomy
TopicsPolynomial and algebraic computation · Mathematical functions and polynomials · Holomorphic and Operator Theory