Math Problems Vault

Problem 6.7.1

Fix a positive real number $a$ and let $\chi_{[-a, a]}$ denote the characteristic function

\chi _{[-a, a]}(t)= \begin{cases} 1 & \text{ if } t \in [-a, a] \\ 0 & \text{ otherwise }\end{cases}

Show that the Fourier transform of $\chi_{[-a, a]}$ is given by

\widehat{\chi }_{[-a, a]}(w)=\frac{2 \sin (a w)}{w}

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Problem 6.7.2

As in Example 1, fix real numbers $\theta$ and $a$ , and let

f(t)= \begin{cases} \frac{1}{\sqrt{2 a}} e^{i \theta t} & \text{ if } t \in (-a, a) \\ 0 & \text{ otherwise. }\end{cases}

Show that $\int_{-\infty }^{\infty }|f(t)|^{2} d t=1$ and

\hat{f}(w)=\frac{1}{\sqrt{\pi a}} \frac{\sin a(w-\theta )}{w-\theta }

Use that $e^{i \theta }=\cos \theta +i \sin \theta$ for every real number $\theta$ .

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Problem 6.7.3

In the section we claimed that

\sigma _{w}^{2}=\int _{-\infty }^{\infty } w^{2}|\hat{f}(w)|^{2} d w=\int _{-\infty }^{\infty }|\widehat{D f}(w)|^{2} d w=\left\| D f\right\| ^{2}

The point of this exercise is to verify the middle equality, and hence all three equalities. Note that

\widehat{D f}(w)=\frac{1}{\sqrt{2 \pi }} \int _{-\infty }^{\infty } f^{\prime }(t) e^{-i w t} d t.

Use integration by parts to show that this equals $i w \hat{f}(w)$ , and hence that the equation holds.

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Problem 6.7.4

Verify the formula

T f(t)=\int _{-a}^{a} \frac{\sin b(t-s)}{\pi (t-s)} f(s) d s

for $T=T_{b \ell } T_{t \ell }$ , as used in the section.

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Problem 6.7.5

Show that (6.8) (in the text of the section) is true whenever $\alpha^{2}+\beta^{2} \leq 1$ .

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