Chapter 3

Calculate the Fourier transform of $$ f(x)= \begin{cases}1-x^{2}, & |x| \leq 1 \\ 0, & |x|>1\end{cases} $$ and show that $$ \int_{0}^{\infty}\left(\frac{x \cos x-\sin x}{x^{3}}\right) \cos \frac{x}{2} d x=\frac{3 \pi}{16} $$

Determine the Fourier transform of each of the following functions: (a) \(f(x)=\frac{1}{a^{2}+x^{2}}\) (b) \(f(x)=\frac{\cos a x}{a^{2}+x^{2}}\) (c) \(f(x)=\frac{\sin b x}{a^{2}+x^{2}}\)

Prove that \(I=\int_{-\infty}^{\infty} e^{-x^{2}} d x=\sqrt{\pi}\). (Hint: It is not difficult to show that \(I^{2}=\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{-\left(x^{2}+y^{2}\right)} d x d y\). By a change of variables to polar coordinates prove that \(I^{2}=\pi\).)

For each \(a>0\) we define the functions $$ f_{a}(x)=\left\\{\begin{array}{ll} 1, & |x|

For each \(x>0\), we define \(f(x)=e^{-x} \cos x .\) Let \(\tilde{f}\) be the odd continuation of \(f\). Prove that for all \(x \neq 0\), $$ \frac{2}{\pi} \int_{0}^{\infty} \frac{t^{3} \sin x t}{t^{4}+4} d t=\tilde{f}(x) . $$

Let $$ H(x)= \begin{cases}0, & x<0, \\ 1, & x \geq 0 .\end{cases} $$ Find the Fourier transform of (a) \(f(x)=H(x) e^{-a x}, \quad a>0\) (b) \(f(x)=H(x) e^{-a x} \cos b t, \quad a>0, \quad b \neq 0\) (c) \(f(x)=H(x) e^{-a x} \sin b t, \quad a>0, \quad b \neq 0\)

Determine the Fourier transform of $$ f(x)= \begin{cases}1, & 00\), and use it to calculate $$ \int_{0}^{\infty} \frac{\cos a x-1}{x} \sin b x d x, \quad b>0 . $$

Let \(f \in G(\mathbb{R})\). Determine the Fourier transform of (a) \(f(-x)\) (b) \(f\left(x-x_{0}\right), x_{0}\) is a real constant (c) \(f(x) e^{i \omega_{0} x}, \omega_{0}\) is a real constant (d) \(f(x) \sin \omega_{0} x\) (e) \(f(x) \cos \omega_{0} x\) (f) \(e^{i x} f(3 x)\) (g) \(f(2 x)\)

Let $$ f(x)= \begin{cases}e^{-x}, & x>0, \\ 0, & x \leq 0 .\end{cases} $$ (a) Calculate the Fourier transform \(F\) of \(f\). (b) Determine \(f * f\) and \((f * f) *(f * f)\). (c) Find \(\mathcal{F}[(f * f) *(f * f)]\). (d) Calculate the integral \(\int_{-\infty}^{\infty} \frac{1}{\left(1+x^{2}\right)^{4}} d x\).