Math205b Winter Term 2003

Assignment 14

1. Let the function f ∈ C(ℝ, ℝ) be given by

f(x) = \begin{cases}1 − |x| , & |x| < 1 , \\ 0 , & |x| ≥ 1 . \end{cases}

Compute f ∗ f and f ∗ f ∗ f.

2. Show that f : ℝ → ℝ , x ↦ \frac{1}{1+x^{2}} is analytic.

3. Let f ∈ C_{c}(ℝ, ℝ) and g ∈ C^{1}(ℝ, ℝ). Show that f ∗ g ∈ C^{1}(ℝ, ℝ) .

4. For f ∈ C_{c}(ℝ, ℝ) define its Fourier transform f̂: ℝ → 𝕂 by

f̂(ξ) = ∫_{−∞}^{∞} e^{−ixξ}f(x) dx .

Show that f̂ is well-defined and analytic. Give an estimate for the radius of convergence of its power series expansion about ξ = 0.

5. You ask a question.

The Homework is due on Friday, February 28.