Polynomials and Trigonometric Polynomials

Abstract

Setting cos ϑ=x, the expressions

$$ T_n (x) = \cos n\vartheta ,\quad U_n (x) = \frac{1} {{n + 1}}T_{n + 1}^\prime (x) = \frac{{\sin (n + 1)\vartheta }} {{\sin \vartheta }},\quad n = 0,1,2 \ldots $$

are polynomials in x of degree n (the Tchebychev polynomials); the leading coefficient of T _n (x) is equal to 2ⁿ⁻¹, and that of U _n (x) is equal to 2ⁿ, n=1,2,3,...