I have been lastly working with transformations of the following type: \begin{eqnarray} f(t)=\sum_{k}\int_{-a}^{a}dx e^{ixt+ikb}p_{k}(x,t)g_{k}(x) \end{eqnarray} By looking around, I discovered something known as "pseudo-differential operators", which looks a bit like this but without the discrete sum. My problem basically deals with the calculation of the $p_{k}(x,t)$ coefficients, which I know must satisfy certain constrains and are just complex functions, possibly without any singularities. For example, one of the conditions is required that:
\begin{eqnarray} \int_{-\infty}^{\infty} dt e^{ixt+ikb-ix't-ik'b}p_{k'}^{*}(x',t)p_{k}(x,t)=2\pi\delta(x-x')\delta_{kk'} \end{eqnarray} which is orthogonality between the functions. Another important condition is for example to require that: \begin{eqnarray} \int_{-\infty}^{\infty} dt e^{ixt+ikb-ix't-ik'b}p_{k'}^{*}(x',t)\partial_{t}p_{k}(x,t)=0 \end{eqnarray} along with other equations like: \begin{eqnarray} \int_{-\infty}^{\infty} dt e^{ixt+ikb-ix't-ik'b}p_{k'}^{*}(x',t')F(t',t)p_{k}(x,t)=G_{kk'}(x,x') \end{eqnarray}
My question is, what is the general approach in this case in order to find the $p_{k}(x,t)$ functions? That is, from the set of equations above, can one somehow calculate the $p_{k}(x,t)$ in an exact way by some inversion method ? I am pretty sure there is a general theory for solving these problems, I just don't know where to start from. I am a bit familiar with Fredholm integral equations, but not so much in the case when there are both discrete and continuous indices involves. I would be glad if someone could shed some light into this and what is the correct path to follow in order to make the transformation exact, thank you