Chebyshev Polynomials

Manuel Eberl 📧

November 13, 2023

This is a development version of this entry. It might change over time and is not stable. Please refer to release versions for citations.


The multiple-angle formulas for $\cos$ and $\sin$ state that for any natural number $n$, the values of $\cos nx$ and $\sin nx$ can be expressed in terms of $\cos x$ and $\sin x$. To be more precise, there are polynomials $T_n$ and $U_n$ such that $\cos nx = T_n(\cos x)$ and $\sin nx = U_n(\cos x)\sin x$. These are called the Chebyshev polynomials of the first and second kind, respectively.

This entry contains a definition of these two familes of polynomials in Isabelle/HOL along with some of their most important properties. In particular, it is shown that $T_n$ and $U_n$ are orthogonal families of polynomials.

Moreover, we show the well-known result that for any monic polynomial $p$ of degree $n > 0$, it holds that $\sup_{x\in[-1,1]} |p(x)| \geq 2^{n-1}$, and that this inequality is sharp since equality holds with $p = 2^{1-n}\, T_n$. This has important consequences in the theory of function interpolation, since it implies that the roots of $T_n$ (also colled the Chebyshev nodes) are exceptionally well-suited as interpolation nodes.


BSD License


Session Chebyshev_Polynomials