Overview
Problem 1. Suppose f : R ! R is a continuous function which can be uniformly approximated by
polynomials on R. Show that f is itself a polynomial.
(Hint: If Pn and Pm are polynomials, then so is Pn Pm. Assuming jPn(x) Pm(x)j < " for all x 2 R,
what does that tell you about Pn Pm? Sub-hint: how do polynomials behave at in nity?)
Remark. Note that this exercise implies that the Weierstrass approximation theorem fails in general when
we are not dealing with a compact domain.
Problem 2. Consider the space C(R;R) of continuous real-valued functions.
(a) Given any f 2 C(R;R), show that there exists a sequence of polynomials (Pn)n that converges
pointwise to f, i.e. limn!1 Pn(x) = f(x) for every x 2 R.
(b) Show that one can moreover choose the sequence (Pn)n in such a way that the convergence is uniform
on any compact subset of R, i.e. for every K R compact, Pn ! f uniformly on K. (Note: the
sequence Pn does not depend on K!)
Problem 3. Exercise 4.1.2 from the textbook. (Note that we saw most examples you will need in class.
Explain brie
y for every example why it satis es the requirements.)
Problem 4. Show that the function f : R n f1g ! R given by f(x) = 1
1x is real analytic on its domain.
Given a 6= 1, what is the radius of convergence of the power series of 1
1x centered at a?
Problem 5. Discuss the convergence of the following series by calculating the radius of convergence R
and checking what happens at the \boundary points" x = R.
1X
n=1
(1)n+1
n
xn and
1X
n=0
1
3nn2 xn:
Do these series converge uniformly on any of the intervals [r;R]; [R; r], or [R;R] where 0 < r < R?
Problem 6. Let a > 0. As another application of the contraction mapping theorem, we will nd an
exponentially fast algorithm to calculate
p
a: de ne
f(x) =
1
2
x +
a
x
:
(a) Show that
p
a is a xed point of f.
(b) Show that f([
p
a;1)) [
p
a;1).
(c) Show that f is a contraction on [
p
a;1). Conclude that starting with any x0 2 [
p
a;1), the sequence
(xn)n de ned by xn = f(xn1) converges to
p
a.
(Hint: you can use the following fact, which follows easily from the fundamental theorem of calculus.
For f di erentiable:
jf(x) f(y)j
max
xy
f0()
jx yj :
Optional: Prove this!)
(d) Take a = 2, x0 = 2 and compute xn for n = 1; 2; 3.