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What is Chapter 6. Weierstrass Approximation Theorem about?

Every continuous function on a bounded interval can be approximated to arbitrary accuracy by polynomials. This is the famous Weierstrass approximation theorem, proved by Karl Weierstrass when he was 70 years old [Weierstrass 1885]. The theorem was independently discovered at about the same time, in essence, by Carl Runge: as pointed out in 1886 by Phragmén in remarks published as a footnote stretching over four pages in a paper by Mittag-Leffler [1900], it can be derived as a corollary of results Runge published in a pair of papers in 1885 [Runge 1885a[Runge & 1885b]].Here and throughout this book, unless indicated otherwise, • denotes the supremum norm on [−1, 1]. Theorem 6.1. Weierstrass approximation theorem. Let f be a continuous function on [−1, 1], and let ε > 0 be arbitrary. Then there exists a polynomial p such that f − p < ε.

Published

2019

Language

EN