Polynomials and Trigonometric Polynomials

Abstract

Setting cos ϑ = x, the expressions

$$ T_n \left( x \right) = \cos n\vartheta {\text{ }}U_n \left( x \right) = \frac{1} {{n + 1}}T'_{n + 1} \left( x \right) = \frac{{\sin \left( {n + 1} \right)\vartheta }} {{\sin \vartheta }}'{\text{ }}n = 0,1,2,... $$

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