Chapter 3

Let $$ f(x)= \begin{cases}x-a, & a0\), and let \(F(\omega)\) be the Fourier transform of \(f\). Let \(g \in G(\mathbb{R})\) be a continuous function whose Fourier transform \(G(\omega)\) satisfies the equality \(G(\omega)=2 \pi F(\omega) F(-\omega)\). Calculate \(g(2 a)\).

Prove that for every \(a>b>0\) the following inequality holds: $$ \int_{0}^{\infty} \frac{\cos a x}{x^{2}+b^{2}} d x>\int_{0}^{\infty} \frac{\cos b x}{x^{2}+a^{2}} d x . $$

Given \(\mathcal{F}[f](\omega)=\frac{1}{1+\omega^{2}}\), calculate \(\mathcal{F}\left[x^{2} f^{\prime \prime}(x)+2 f^{\prime \prime \prime}(x)\right](\omega)\).

Solve the following boundary-value problem: $$ \begin{cases}u_{t}=4 u_{x x}, & -\infty2 .\end{cases} $$

Find all values \(\omega\) for which \(\int_{-\infty}^{\infty} \frac{\sin ^{2} 5 x}{x^{2}} \cos \omega x d x=0\)

Let \(f \in C\left(\mathbb{R}^{n}\right)\) be absolutely integrable on \(\mathbb{R}^{n} .\) For real \(\omega_{1}, \omega_{2}, \ldots, \omega_{n}\) we define \(\mathcal{F}[f]\left(\omega_{1}, \ldots, \omega_{n}\right)=\) $$ \frac{1}{(2 \pi)^{n}} \underbrace{\int_{-\pi}^{\pi} \cdots \int_{-\pi}^{\pi}}_{n \text { times }} f\left(x_{1}, \ldots, x_{n}\right) e^{-i\left(\omega_{1} x_{1}+\cdots+\omega_{n} x_{n}\right)} d x_{1} \cdots d x_{n} . $$ The function \(\mathcal{F}[f]\) (defined on \(\mathbb{R}^{n}\) ) is said to be the multivariate Fourier transform of \(f\). Show that if \(f_{1}, f_{2}, \ldots, f_{n}\) are univariate continuous functions in \(G(\mathbb{R})\), and if $$ f\left(x_{1}, x_{2}, \ldots, x_{n}\right)=f_{1}\left(x_{1}\right) f_{2}\left(x_{2}\right) \cdots f_{n}\left(x_{n}\right) $$ then $$ \mathcal{F}[f]\left(\omega_{1}, \ldots, \omega_{n}\right)=\mathcal{F}\left[f_{1}\right]\left(\omega_{1}\right) \mathcal{F}\left[f_{2}\right]\left(\omega_{2}\right) \cdots \mathcal{F}\left[f_{n}\right]\left(\omega_{n}\right) $$