Chapter 3

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 \(\int_{-\infty}^{\infty} \frac{1}{\left(1+x^{2}\right)^{4}} d x\).

Compute the Fourier transform of \(f(x)=\frac{1}{e^{x}+e^{-x}}\). (Hint: Use the simple closed curve \(\gamma_{R}=I_{R} \cup J_{R} \cup I_{R}^{\prime} \cup J_{R}^{\prime}\), where $$ \begin{aligned} &I_{R}=\\{z \mid z=x, \quad-R \leq x \leq R\\}, \\ &J_{R}=\\{z \mid z=R+i y, \quad 0 \leq y \leq \pi\\}, \\ &I_{R}^{\prime}=\\{z \mid z=x+i \pi, \quad-R \leq x \leq R\\}, \\ &J_{R}^{\prime}=\\{z \mid z=-R+i y, \quad 0 \leq y \leq \pi\\}, \end{aligned} $$ with positive orientation, and let \(R \rightarrow \infty\).)

Assume that \(f\) and \(f^{\prime}\) are continuous, \(f, f^{\prime}, f^{\prime \prime} \in G(\mathbb{R})\), and \(x f(x)\) is also absolutely integrable. Assume that \(f\) satisfies the differential equation $$ f^{\prime \prime}(x)+2 x f^{\prime}(x)+2 f(x)=0 . $$ What differential equation does \(\mathcal{F}[f]\) satisfy?

Let \(f: \mathbb{R} \rightarrow \mathbb{C}\) be a continuous, absolutely integrable function. Let \(F\) denote the Fourier transform of \(f\). Show that if \(F(\omega)=0\) for all \(\omega>\left|\omega_{0}\right|\) then for all \(a>\left|\omega_{0}\right|\) $$ f(x) * \frac{\sin a x}{\pi x}=f(x) . $$

Using the generalized Plancherel identity, calculate the integral $$ \int_{0}^{\infty} \frac{d t}{\left(a^{2}+t^{2}\right)\left(b^{2}+t^{2}\right)}, \quad a, b>0 . $$

Assume that \(f\) and \(f^{\prime}\) are continuous, \(f, f^{\prime}, f^{\prime \prime} \in G(\mathbb{R})\), and \(x f(x)\) is also absolutely integrable. Use Fourier transforms to solve the equation $$ f^{\prime \prime}(x)+x f^{\prime}(x)+f(x)=0, \quad f(0)=1, \quad f^{\prime}(0)=0 . $$

Let \(F\) denote the Fourier transform of the function $$ f(x)= \begin{cases}1, & 0 \leq x<1 \\ 0, & \text { otherwise }\end{cases} $$ Find the function \(g\) such that the Fourier transform \(G\) of \(g\) satisfies \(G(\omega)=\) \([F(\omega)]^{2} .\)

For each \(a>0\), let \(f_{a}(x)=e^{-a|x|}\), and \(g_{a}(x)=\frac{2 a}{x^{2}+a^{2}}\). (a) Find the Fourier transform of \(f_{a}\). (b) Find the Fourier transform of \(g_{a}\). (c) Does there exist a function \(\varphi \in G(\mathbb{R})\) such that \(\int_{-\infty}^{\infty} \frac{\varphi(t)}{(x-t)^{2}+16} d t=\) \(\frac{1}{x^{2}+49}\) ? If yes, find it. If no, explain why it cannot exist. (d) Does there exist a function \(\varphi \in G(\mathbb{R})\) such that \(\int_{-\infty}^{\infty} \frac{\varphi(t)}{(x-t)^{2}+49} d t=\) \(\frac{1}{x^{2}+16}\) ? If yes, find it. If no, explain why it cannot exist.

Let \(f: \mathbb{R} \rightarrow \mathbb{C}\) be continuous and absolutely integrable on \(\mathbb{R}\). Let \(F\) be the Fourier transform of \(f\). It is known that $$ F(\omega)= \begin{cases}1-\omega^{2}, & |\omega| \leq 1 \\ 0, & |\omega|>1\end{cases} $$ Find \(f\).

Let \(f\) be a function which is twice differentiable on all of \(\mathbb{R}\) such that \(f(t)\), \(f^{\prime}(t), f^{\prime \prime}(t), t f(t)\), and \(t^{2} f(t)\) are continuous and absolutely integrable over \(\mathbb{R}\). We denote the Fourier transform of \(f\) by \(F\). Find a real number \(c\) such that if $$ f^{\prime \prime}(t)+\left(t^{2}-2\right) f(t)=c f(t) $$ then $$ F^{\prime \prime}(\omega)+\left(\omega^{2}-2\right) F(\omega)=c F(\omega) . $$