WA 2
April 8, 2009
Submit your solutions on Tuesday, April 14.
1. (2 pt.) Problem 1.11 (the case |α|<1)
Hint: Use an argument similar to the one in Example 1.16.
2. (an extra credit problem, 1 pt.) Problem 1.11 (the case |α|>1)
Hint: Use the Cauchy–Goursat theorem from Complex Analysis.
3. (2 pts.) Problem 2.3
4. (an extra credit problem, 2 pts.) Problem 2.6
Hint: You might find useful the fact that if a sequence of functions converges
in the supremum-norm then it converges to the same limit function in the norm
induced by the inner product (prove it if you use it).
5. (2 pts.) Problem 2.14
Hint: In order to prove that the subspace RH2is closed in RL2it suffices to
show that for any sequence of functions fn∈RH2which converges in RL2, the
limit function fbelongs to RH2. Let {fn}be such a convergent sequence, and let
the limit function (which is rational by the definition of RL2) be represented as
f(z) = p(z)/q(z) where p(z) and q(z) are co-prime polynomials (i.e., they have no
common zeros).
Step 1. Show that fnq∈RH2for n= 1,2, . . .,p∈RH 2, and fnq→pin RH 2
as n→ ∞.
Step 2. Use the Szeg¨o reproducing kernel kα(z) (see Problem 1.11) to show that
for any α∈D,fn(α)q(α)→p(α) as n→ ∞.
Step 3. Use the result of Step 2 to show that q(α)6= 0 for any α∈D.
Step 4. Conclude from Step 3 that f∈RH2.
6. (2 pts.) Problem 3.2
Hint: Follow the outline that was given in class.
7. (2 pts.) Show that (C[0,1],k·k∞) is a Banach space.
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