A trace Paley-Wiener theorem for $\mathrm{GL}_{n}(F)\backslash \mathrm{GL}_{n}(E)$, with $E/F$ a quadratic extension of non-Archimedean local fields

We extend the classical trace Paley-Wiener theorem for $p$-adic reductive groups, given by Bernstein, Deligne and Kazhdan in 1986, to a relative case. That is when $E/F$ is a quadratic extension of $p$-adic fields, the main result is a scalar Paley-Wiener theorem for relative Bessel distributions on