2015-47.md

course	course_year	question_number	tags	title	year
Methods	IB	47	IB 2015 Methods	Paper 4, Section I, 5C	2015

(a) The convolution $f * g$ of two functions $f, g: \mathbb{R} \rightarrow \mathbb{C}$ is related to their Fourier transforms $\tilde{f}, \tilde{g}$ by

$$\frac{1}{2 \pi} \int_{-\infty}^{\infty} \tilde{f}(k) \tilde{g}(k) e^{i k x} d k=\int_{-\infty}^{\infty} f(u) g(x-u) d u$$

Derive Parseval's theorem for Fourier transforms from this relation.

(b) Let $a>0$ and

$$f(x)= \begin{cases}\cos x & \text { for } x \in[-a, a] \ 0 & \text { elsewhere }\end{cases}$$

(i) Calculate the Fourier transform $\tilde{f}(k)$ of $f(x)$.

(ii) Determine how the behaviour of $\tilde{f}(k)$ in the limit $|k| \rightarrow \infty$ depends on the value of $a$. Briefly interpret the result.