9.4 *Two Results of Weierstrass 577Show that Pn(x) is a polynomial in x^2 of degree 2n-^1.
Use mathematical induction to prove that Vx E [-1, 1], 0::; Pn(x) ::; l.
Show that Vx E [-1, 1], {pn(x)} is monotone decreasing.
Therefore, 'ix E [-1, 1], {pn(x)} converges (why?).
Define the function L : [-1, 1] ---+ JR by L( x) = lim Pn ( x), and prove that
n--><Xl
L(x) = 1 - JxJ.
Show that Dini's uniform convergence theorem^16 (9.3.14) guarantees that
Pn(x) converges uniformly to 1 - JxJ on [-1, l]. •
Corollary 9.4.14 The function JxJ can be approximated by polynomials on
[-1, l].Proof. Apply Lemma 9.4.5 and Theorem 9.4.13. •Corollary 9.4.15 Given any c E JR, the function Jx - cJ can be approximated
by polynomials on any compact interval [a, b].Proof. Suppose a < b and c E R Choose any d > 0 such that [a, b] ~
[ c -d, c + d]. Let { qn ( x)} be a sequence of polynomials converging uniformly
to JxJ on [-1, 1] guaranteed by Corollary 9.4.14.
x-c
Now V x E [c - d, c + d], let t = -d-; show that -1 ::; t::; l.Definethepolynomialsrn(x) on [c-d,c+d] byrn(x) = dqn (x;tc) = dqn(t).
Show that {rn} converges uniformly to Jx - cJ on [c - d, c + d], and hence on
[a,b]. •
Corollary 9.4.16 Ve E JR, the function (x - c)+ can be approximated by poly-
nomials on any compact interval [a, b].Corollary 9.4.17 Every polygonal f : [a, b] ---+JR can be approximated by poly-
nomials on [a, b].Theorem 9.4.18 (Weierstrass' Approximation Theorem) Every con-
tinuous function f : [a, b] ---+ JR can be approximated by polynomials on [a, b].
Corollary 9.4.19 Given any continuous f : [a, b] ---+ JR, there is a sequence of
polynomials converging uniformly to f on [a, b].16. I bet you thought you would never use that theorem!