1549901369-Elements_of_Real_Analysis__Denlinger_

576 Chapter 9 • Sequences and Series of Functions

n

(c) 'tit E [xi-1, xi], f(t) = f(a) + 2.: (mi - mi-1)(t - Xi-1)+.

i=l

Since this formula applies in every subinterval [xi-l, xi], we conclude that

n

'tit E [a, b], f(t) = f(a) + 2.: (mi - mi-1)(t - Xi-1)+. •

i=l

We now need a few more technical results that will lead us to a proof of
Weierstrass' theorem.

Lemma 9.4.12 Ve E JR, the function (x - c)+ is expressible in the form

(x _ c)+ = (x - c) + Ix - cl

2.

This suggests that we should now focus our attention on approximating

Ix - cl by polynomials.

Theorem 9.4.13 The function f(x) = 1-lxl can be approximated by polyno-
mials on [-1, l].

y

Figure 9.15

Proof. We shall use Lemma 9.4.3. Let y = 1 - lxl on [-1, 1]. Show that
this is equivalent to
x^2 = 1 - 2y + y^2 , 0 :::; y :::; 1,
which is equivalent to
y2 + (1 - x2)
y=--~-~ 2.

Define a sequence of polynomials {Pn} recursively by

{

Po(x) = 1, and 'tin EN,

( )

P~-l (x) + (1 - x^2 )

Pn x =

2

.

Write out the first three polynomials in this sequence to see what they look
like.