\documentclass{article}
\usepackage{amsmath}
\begin{document}

\noindent
Substitutions:
\[
  \begin{split}
    \phi &\leftarrow \psi + \frac \theta 2
    \\
    \frac \theta 2 &= \phi - \psi
    \\
    \theta &= 2\phi - 2\psi
    \\
    d\theta &= 2d\phi
    \\
    \psi - \frac \theta 2 &= \psi - (\phi - \psi) = 2\psi - \phi
    \\
    \chi &\leftarrow \phi-2\psi
    \\
    d\chi &= -2\,d\psi
  \end{split}
\]
Calculations:
\[
  \begin{split}
    \lefteqn{\mathcal F \int h\left(\psi - \frac v 2\right)
      f\left(\psi + \frac v 2\right)d\psi} &
    \\&=
      \frac 1{\sqrt{2\pi}}
      \int e^{-iv\theta}
        \int h\left(\psi - \frac \theta 2\right)
        f\left(\psi + \frac \theta 2\right)d\psi\,d\theta
    \\&=
      \frac 1{\sqrt{2\pi}}
      \int\int e^{-iv\theta}
        h\left(\psi - \frac \theta 2\right)
        f\left(\psi + \frac \theta 2\right) d\theta\,d\psi
    \\&=
      \frac 2{\sqrt{2\pi}}
      \int e^{-iv(2\phi-2\psi)}
        h\left(2\psi - \phi\right)
        f\left(\phi\right) d\phi\,d\psi
    \\&=
      \frac 2{\sqrt{2\pi}}
      \int e^{-iv\phi}
        f\left(\phi\right)
      \int e^{-iv(\phi-2\psi)}
        h\left(2\psi - \phi\right)
        d\psi\,d\phi
    \\&=
      \frac 1{\sqrt{2\pi}}
      \int e^{-iv\phi}
        f\left(\phi\right)
      \int e^{-iv\chi}
        h\left(- \chi \right)
        d\chi\,d\phi
    \\&=
      \frac 1{\sqrt{2\pi}}
      \int e^{-iv\chi}
        h\left(- \chi \right)
        d\chi
      \int e^{-iv\phi}
        f\left(\phi\right)
        d\phi
  \end{split}
\]

\newpage
\noindent
Skew:
\[
  \begin{array}{ccccccccc}
    \rightarrow &
    \Phi(-w) &
    \rightarrow &
    \phi(w) &
    \rightarrow &
    \Phi(w) &
    \rightarrow &
    \phi(-w) &
    \rightarrow \\
    \rightarrow &
    f(w) &
    \rightarrow &
    F(w) &
    \rightarrow &
    f(-w) &
    \rightarrow &
    F(-w) &
    \rightarrow \\
    \rightarrow &
    \Sigma(-w) &
    \rightarrow &
    \sigma(w) &
    \rightarrow &
    \Sigma(w) &
    \rightarrow &
    \sigma(-w) &
    \rightarrow \\
    \rightarrow &
    h(w) &
    \rightarrow &
    H(w) &
    \rightarrow &
    h(-w) &
    \rightarrow &
    H(-w) &
    \rightarrow
  \end{array}
\]
Skew:
\[
  \begin{array}{ccccccccc}
    \rightarrow &
    \Gamma(-v) &
    \rightarrow &
    \gamma(v) &
    \rightarrow &
    \Gamma(v) &
    \rightarrow &
    \gamma(-v) &
    \rightarrow \\
    \rightarrow &
    g(v) &
    \rightarrow &
    G(v) &
    \rightarrow &
    g(-v) &
    \rightarrow &
    G(-v) &
    \rightarrow
  \end{array}
\]
Calculations:
\[
  \begin{split}
    \gamma(v) &\equiv \int \sigma\left(\psi - \frac v 2\right)
      \phi\left(\psi + \frac v 2\right)d\psi \\
    \Gamma(v) &= \sqrt{2\pi}\Sigma(-v)\Phi(v) \\
    \gamma(-v) &= \int \sigma\left(\psi + \frac v 2\right)
      \phi\left(\psi - \frac v 2\right)d\psi \\
    \Gamma(-v) &= \sqrt{2\pi}\Sigma(v)\Phi(-v) \\
    G(v) &= \int H\left(\psi - \frac v 2\right)
      F\left(\psi + \frac v 2\right)d\psi \\
    g(-v) &= \sqrt{2\pi}h(v)f(-v) \\
    G(-v) &= \int H\left(\psi + \frac v 2\right)
      F\left(\psi - \frac v 2\right)d\psi \\
    g(v) &= \sqrt{2\pi}h(-v)f(v)
  \end{split}
\]
Conclusion:
\[
  \begin{split}
    g(v) &\rightarrow G(v) \\
    h(-v)f(v) &\rightarrow \frac 1 {\sqrt{2\pi}}
      \int H\left(\psi - \frac v 2\right)
      F\left(\psi + \frac v 2\right)d\psi
  \end{split}
\]

\end{document}
