I have an integral equation involving an unknown function $f(x)$, of the most basic form
$$ \int_{-1}^{1} e^{iω(t) x} f(x) \ dx = g(t) $$
I am solving for an approximation of $f(x)$ by substituting in its truncated Legendre expansion, $f(x) \approx \sum^N a_j P_j(x)$. Then, the integral equation reduces to a product of the sum coefficients $a_j$ and the Fourier transforms of the Legendre polynomials (expressed in closed form using Bessel functions). That is,
$$ \sum^N a_j \hat P_j(ω(t)) = g(t) $$
The sum coefficients are found by collocating in the free variable $t$ and solving the resulting linear system.
Since $f(x)$ is unknown, how can I investigate the accuracy of the computed $a_j$ versus their actual value, and also the rate of convergence? This method is unlike a traditional finite difference method, or traditional Legendre interpolation, so I am unsure of what kind of literature to look for.
Edit for extra info: Since the coefficients themselves are also approximate, the final computed approximation for $f(x)$ is $\sum^N \tilde a_j P_j(x)$, where $\tilde a_j$ are approximated values of $a_j$. So, in a sense, there are 2 approximations happening here, one for the truncation of the sum and one for the evaluation of the sum coefficients. Which one is more important to study?
Thanks in advance.