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# Approximating Continuous Functions by Polynomials

DOI link for Approximating Continuous Functions by Polynomials

Approximating Continuous Functions by Polynomials book

# Approximating Continuous Functions by Polynomials

DOI link for Approximating Continuous Functions by Polynomials

Approximating Continuous Functions by Polynomials book

## ABSTRACT

The aim of this article is to discuss some ideas relating to the following basic question: How well can we approximate a continuous function on an interval of the real line by simpler functions? This question has given rise to a ﬂourishing area of research. Perhaps the best-known answer to this question (and there is more than one answer to this question depending on the precision of the answer one seeks and on the application that motivated the question) is the Weierstrass Approximation Theorem. It says:

Theorem 1. (Weierstrass, 1885) Let f be a continuous real-valued func-

tion deﬁned on a closed interval [a, b], and let ε > 0 be given. Then, there exists a polynomial p with real coeﬃcients such that | f (x) − p(x)| < ε for every x ∈ [a, b].