If we use Fourier Transform, we can switch from the position representation to the momentum representation, like the following equation

\(\phi(p)=\frac{1}{\sqrt{2\pi\hbar}}\int \psi(x)e^{-ipx/\hbar}dx\)

here comes the problem, if we use dirac notation we can see it is the inverse Fourier Transform we use to switch into the momentum representation

because of \(\langle x|p\rangle=\frac{1}{\sqrt{2\pi \hbar}}e^{\frac{i}{\hbar}px}\),we can get \(\begin{split} |p\rangle=&|x\rangle\langle x|p\rangle=\frac{1}{\sqrt{2\pi \hbar}}e^{\frac{i}{\hbar}px}|x\rangle=\mathscr{F} ^{-1}|x\rangle\\ &\mathscr{F} ^{-1}=\mathscr{F} ^{-1}|x\rangle\langle x|=|p\rangle\langle x| \end{split}\)

This is used in one published paper. title:" A finite-dimensional quantum model for the stock market" author:Liviu-Adrian Cotfas, page 7, formula 32

also how can I get the following relation satisfied by the position and momentum operators ?

\(\hat{p}=\mathscr{F} ^{-1} \hat{x} \mathscr{F}\)

many thanks!