Problems to Sections 5.1, 5.2

$\renewcommand{\Re}{\operatorname{Re}}$ $\renewcommand{\Im}{\operatorname{Im}}$ $\newcommand{\erf}{\operatorname{erf}}$ $\newcommand{\dag}{\dagger}$ $\newcommand{\const}{\mathrm{const}}$ $\newcommand{\arcsinh}{\operatorname{arcsinh}}$

Problems to Sections 5.1, 5.2

  1. Problem 1
  2. Problem 2
  3. Problem 3
  4. Problem 4
  5. Problem 5
  6. Problem 6
  7. Problem 7

Some of the problems could be solved based on the other problems and properties of Fourier transform (see Section 5.2) and such solutions are much shorter than from the scratch; seeing and exploiting connections is a plus.

Problem 1. Let $F$ be an operator of Fourier transform: $f(x)\to \hat{f}(k)$. Prove that

  1. $F^*F=FF^*=I$;
  2. $(F^2 f)(x)=f(-x)$ and therefore $F^2f=f$ for even function $f$ and $F^2=-f$ for odd function $f$;
  3. $F^4=I$;
  4. If $f$ is a real-valued function then $\hat{f}$ is real-valued if and only if $f$ is even and $i\hat{f}$ is real-valued if and only if $f$ is odd.

Problem 2. Let $\alpha>0$. Find Fourier transforms of

  1. $e^{-\alpha|x|}$;
  2. $e^{-\alpha|x|}\cos(\beta x)$, $e^{-\alpha|x|}\sin (\beta x)$ with $\beta>0$;
  3. $x e^{-\alpha|x|}$ with $\beta>0$;
  4. $xe^{-\alpha|x|}\cos(\beta x)$, $x e^{-\alpha|x|}\sin (\beta x)$ with $\beta>0$.

Problem 3. Let $\alpha>0$. Find Fourier transforms of

  1. $(x^2+\alpha^2)^{-1}$;
  2. $x(x^2+\alpha^2)^{-1}$;
  3. $(x^2+\alpha^2)^{-1}\cos (\beta x)$, $(x^2+\alpha^2)^{-1}\sin(\beta x)$;
  4. $x(x^2+\alpha^2)^{-1}\cos (\beta x)$, $x(x^2+\alpha^2)^{-1}\sin(\beta x)$.

Problem 4. Let $\alpha>0$. Based on Fourier transform of $e^{-\alpha x^2/2}$ find Fourier transforms of

  1. $e^{-\alpha x^2/2}\cos (\beta x)$, $e^{-\alpha x^2/2}\sin (\beta x)$;
  2. $ x e^{-\alpha x^2/2}\cos (\beta x)$, $x e^{-\alpha x^2/2}\sin (\beta x)$.

Problem 5. Find Fourier transforms of

a. $f(x)=\left\{\begin{aligned} & 1&& |x|\le a,\\ & 0 && |x|> a;\end{aligned}\right.$

b. $f(x)=\left\{\begin{aligned} & x && |x|\le a,\\ & 0 && |x|> a;\end{aligned}\right.$

c. Using (a) calculate $\int_{-\infty}^\infty \frac{\sin (x)}{x}\,dx$.

Problem 6.

  1. Prove the same properties as in Problem 1 for multidimensional Fourier tramsform (see Subection 5.2.A.

  2. Prove that $f$ if multidimensional function $f$ has a rotational symmetry (that means $f(Q\mathbf{x})= f(\mathbf{x})$ for all orthogonal transform $Q$) then $\hat{f}$ also has a rotational symmetry (and conversely).

Note. Equivalently $f$ has a rotational symmetry if $f(\mathbf{x})$depend only on $|\mathbf{x}|$.

Problem 7. Find multidimenxional Fourier transforms of

  1. $f(x)=\left\{\begin{aligned} & 1&& |\mathbf{x}|\le a,\\ & 0 && |\mathbf{x}|> a;\end{aligned}\right.$

  2. $f(x)=\left\{\begin{aligned} &a-|\mathbf{x}| &&|\mathbf{x}|\le a,\\ &0 &&|\mathbf{x}|> a, \end{aligned}\right.$;

  3. $f(x)=\left\{\begin{aligned} &a^2-|\mathbf{x}|^2 &&|\mathbf{x}|\le a,\\ &0 &&|\mathbf{x}|> a, \end{aligned}\right.$;

  4. $f(x)=e^{-\alpha |\mathbf{x}|}$.

Hint. Using Problem 6(b) observe that we need to find only $\hat{f}(0,\ldots,0, k)$ and use appropriate coordinate system (polar as $n=2$, or spherical as $n=3$ and so one).

Note. This problem could be solved as $n=2$, $n=3$ or $n\ge 2$ (any).

$\Uparrow$  $\uparrow$  $\Rightarrow$