I need to construct Fourier transform of non-equispaced data.
That is, I have signal $s(t)$, $t\in[0,T]$ sampled at non-equispaced points $t_k$, $k=0...N-1$ with sample values $s_k = s(t_k)$. For Fourier transform I use approximation of integral: $$ \hat S(\omega) = \int\limits_{-\infty}^\infty s(t)e^{-i\omega t}dt \approx \hat S_d(\omega) = \sum_{k=0}^{N-1}s_ke^{-i\omega t_k}\Delta t_k \tag{1} $$ where $\Delta t_k = (t_{k+1}-t_{k-1})/2$. As sampling points in frequency domain I choose $\omega_n = \frac{2\pi}{T}$.
My question is: as I can evaluate (1) for any $\omega$, what is the maximum $\omega$ such $\hat S_d(\omega)$ "adequately" represents $\hat S(\omega)$? What is the maximum of $n$ for $\omega_n$ I can use? For DFT we got Nyquist frequency. Do we have something similar for NDFT? Any references would be appreciated.
Note: I'm aware of such things as NDFT and NFFT. Though formula for NDFT, as presented in most papers, is $$ \hat S_d(\omega_n) = \sum_{k=0}^{N-1}s_ke^{-i\omega_n t_k} $$ I strongly believe that I need to use formula (1) as I'm trying to build a periodogram.
And I'm not interested in fast ways of computing NDFT yet, so I'm not considering NFFT.