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arXiv · 2501.02145

Approximation by polynomials with only real critical points

David L. Bishop
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Abstract

We strengthen the Weierstrass approximation theorem by proving that any real-valued continuous function on an interval $I \subset \mathbb{R}$ can be uniformly approximated by a real-valued polynomial whose only (possibly complex) critical points are contained in $I$. The proof uses a perturbed version of the Chebyshev polynomials and an application of the Brouwer fixed point theorem.

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David L. Bishop. 2025-01-04. Approximation by polynomials with only real critical points. https://doi.org/10.4171/rmi%2F1470

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