Real Analysis II

Instructor: Dmitry Ryabogin

Assignment I.

1. Problem 1. Compute 1_{[0,1]} ∗ 1_{[0,1]}(x), x ∈ R.

2. Problem 2. Let f be a periodic (with period 1) integrable function on the real line,

satisfying ∫_{0}^{1} f(t)dt = 0. Prove that ∀α ∈ (0, 1) there exists a set A_{α} ⊂ R such that

∫_{A_{α}} f(t)dt = 0, and m(A_{α}) = α.

Hint. Where is Fubini? (periodicity is important!)

3. Problem 3. Prove that (f ∗ g) ∗ h = f ∗ (g ∗ h), provided f, g, h are integrable on the real line.

4. Problem 4. Put ϕ(t) = 1 − cos t if 0 ≤ t ≤ 2π, ϕ(t) = 0 for all other real t. For −∞ < x < ∞, define

f(x) = 1,  g(x) = ϕ′(x),  h(x) = ∫_{−∞}^{x} ϕ(t)dt.

Verify the following statements about convolutions of these functions:

(i) f ∗ g(x) = 0 for all x.

(ii) g ∗ h(x) = ϕ ∗ ϕ(x) > 0 on (0, 4π).

(iii) Therefore (f ∗g)∗h = 0, whereas f ∗(g ∗h) is a positive constant. But convolution is supposedly associative, by Fubini Theorem (see Problem 2). What went wrong?

5. Problem 5.

Use Fubini Theorem and the relation

\frac{1}{x} = ∫_{0}^{∞}e^{−xt}dt, x > 0,

to prove that

lim_{A→∞} ∫_{0}^{A} \frac{sin x}{x}dx = \frac{π}{2} .

6. Problem 6. Suppose A is a measurable subset of the real line, having finite positive measure. Show that the convolution 1_{A} ∗ 1_{A} is continuous and not identically 0. Use this to prove that A + A contains a segment.