Mathematics 3MI

Department of Mathematics, I.M.F.

Hans Plesner Jakobsen March 9 2003

Weekly notice #6

The lectures in week 10: We finished the integration of positive functions. The main result is Lebesgue’s Monotone Convergence Theorem (Hovedssætning 4.3) and Fatou’s Lemma (4.7).

The lectures in week 11: We will continue with §4.2 and after that §4.7.

Homework - to be handed in to the TA in week 12. : 3MI -exam January 1999 Problem 2: ii), iii), and iv) and June 1999 Problem 3. (The first problem can be found on the Weekly Notice #4, the second may be found below).

The problem sessions in week 12: 4.8*, 4.19, 4.23, 4.24*, = 4.41*, 4.42, W6.1, and W6.2.

Problem W6.1. Show that

lim_{n→∞} ∫_{0}^{2π} sin(x)^{n}dx = 0.

Problem W6.2. For f ∈ L(ℝ, 𝔹, m), consider the Fourier Transform g of f given by

g(t) = \frac{1}{\sqrt{2π}} ∫_{ℝ} f(x)e^{−i·xt}dx.

Prove that the right hand side makes sense and in fact defines a continuous function of t. Set f_{1}(x) = −i · xf(x). Prove that if f_{1} ∈ L(ℝ, 𝔹, m) then g is differentiable and

g′(t) = \frac{1}{\sqrt{2π}} ∫_{ℝ} f_{1}(x)e^{−i·xt}dx.

June 1999 Problem 3: Set

F (t) = ∫_{0}^{1} sin(x^{2} + t^{2})dx , t ∈ [−10, 10].

Give the reasons why F is continuous and differentiable, and show that F′(t) = ∫_{0}^{1} 2t cos(x^{2} + t^{2})dx.