Problems to Sections 5.1, 5.2

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Problems to Sections 5.1, 5.2

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 the unitary Fourier transform: $f(x)\to \hat{f}(k)$ with a factor $(2\pi)^{-1/2}$. 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 $\hat{f}(k)$ of functions

  1. $f(x)=e^{-\alpha|x|}$;
  2. $f(x)=\left\{\begin{aligned} &e^{-\alpha x}&& x>0,\\ &0 &&x\le 0;\end{aligned}\right.$
  3. $f(x)=\left\{\begin{aligned} &xe^{-\alpha x}&& x>0,\\ &0 &&x\le 0;\end{aligned}\right.$
  4. $f(x)=e^{-\alpha|x|}\cos(\beta x)$;
  5. $f(x)=e^{-\alpha|x|}\sin (\beta x)$;
  6. $f(x)=x e^{-\alpha|x|}$;
  7. $f(x)=xe^{-\alpha|x|}\cos(\beta x)$;
  8. $f(x)=x e^{-\alpha|x|}\sin (\beta x)$;
  9. $f(x)=x^2 e^{-\alpha|x|}$.

Problem 3. Let $\alpha>0$. Find Fourier transforms $\hat{f}(k)$ of functions

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

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

  1. $f(x)=x e^{-\alpha x^2/2}$;
  2. $f(x)=x^2 e^{-\alpha x^2/2}$;
  3. $f(x)= e^{-\alpha x^2/2}\cos (\beta x)$;
  4. $f(x)=e^{-\alpha x^2/2}\sin (\beta x)$;
  5. $f(x)=x e^{-\alpha x^2/2}\cos (\beta x)$;
  6. $f(x)=xe^{-\alpha x^2/2}\sin (\beta x)$.

Problem 5. Let $a>0$. Find Fourier transforms $\hat{f}(k)$ of functions

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

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

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

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

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

  6. Using 1. calculate $\int_{-\infty}^\infty \frac{\sin (x)}{x}\,dx$.

Problem 6. Using Complex Variables class (if you took one) find directly Fourier transforms $\hat{f}(k)$ of functions

  1. $(x^2+a^2)^{-1}$ with $a>0$;
  2. $(x^2+a^2)^{-1}(x^2+b^2)^{-1}$, with $a>0$, $b>0$, $b\ne a$;
  3. $(x^2+a^2)^{-2}$ with $a>0$;
  4. $x(x^2+a^2)^{-2}$ with $a>0$;
  5. $(x^2+a^2)^{-1}(x^2+b^2)^{-1}$, with $a>0$, $b>0$, $b\ne a$;
  6. $x(x^2+a^2)^{-1}(x^2+b^2)^{-1}$, with $a>0$, $b>0$, $b\ne a$.

Problem 7.

  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 8. Find multidimentional Fourier transforms $\hat{f}(\mathbf{k})$ of functions

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

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

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

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

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

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

  7. $f(\mathbf{x})=|\mathbf{x}|^2e^{-\alpha |\mathbf{x}|}$.

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 on).

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

$\Uparrow$  $\uparrow$  $\Rightarrow$