 
Summary: Math 311001 201110
Assignment # 4 (due: March 24th)
1. Prove that usual = Cesaro = Abel, and show that the reverse
implications do not hold.
2. Suppose that f C[0, 1] and that
1
0
f(t) tn
dt = 0, n N.
Show that f(x) = 0 for all x [0, 1] (Hint: Weierstrass)
3. Let A be the uniformly closed subalgebra of Cb(R) generated by eix
.
Let B be the uniformly closed subalgebra of Cb(R) generated by sin x
and cos x. Prove that A = B.
4. Let A be the uniformly closed subalgebra of Cb(R) generated by eix
.
Show that
A = {f Cb(R) : f(t) = f(t + 2) for all t R}.
(Hint: first show that you can consider things in a compact set, and
then use StoneWeierstrass)
