Answer Happy

This question involves a version of Weierstrass' Approximation Theorem for Trigonometric Cosine Polynomials. A linear combination of 1, cost, cos 2t, ..., cos nt is called a cosine polynomial of degree ≤ n: n R (t) = a; cos jt j=0 where ao, a₁, ..., an are real numbers. (a) Show by induction on n, that (cost)" is a cosine polynomial of degree <n. (b) Hence show that if P (x) is a polynomial of degree ≤ n in x, then P (cost) is a cosine polynomial of degree ≤n in t. (c) Let h: [0, π] → R be continuous. Let ɛ > 0. Show that there is a cosine polynomial R (t) such that |h (0) − R (0)| < ɛ for all 0 € [0, π] .