Problems and Solutions from Complex Analysis and Functional Analysis, Study notes of Mathematics

A list of problems and solutions from complex analysis and functional analysis, including hints for each problem. Topics covered include convergence in supremum-norm, closed subspaces, and the szegő reproducing kernel. Students are encouraged to use the hints and prove the stated facts to solve the problems.

Typology: Study notes

Pre 2010

Uploaded on 08/19/2009

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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 fnRH2which 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 fnqRH2for n= 1,2, . . .,pRH 2, and fnqpin RH 2
as n .
Step 2. Use the Szeg¨o reproducing kernel kα(z) (see Problem 1.11) to show that
for any αD,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 αD.
Step 4. Conclude from Step 3 that fRH2.
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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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.
  4. (an extra credit problem, 2 pts.) Problem 2. 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. Hint: In order to prove that the subspace RH^2 is closed in RL^2 it suffices to show that for any sequence of functions fn ∈ RH^2 which converges in RL^2 , the limit function f belongs to RH^2. Let {fn} be such a convergent sequence, and let the limit function (which is rational by the definition of RL^2 ) 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 ∈ RH^2 for n = 1, 2 ,.. ., p ∈ RH^2 , and fnq → p in 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 ∈ RH^2.
  6. (2 pts.) Problem 3. Hint: Follow the outline that was given in class.
  7. (2 pts.) Show that (C[0, 1], ‖ · ‖∞) is a Banach space.

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