Shift sampling theory of Fourier transform computation (Q1368237)

The authors consider a modified discrete Fourier transform \((\text{DFT}_{\xi\eta})\) \[ X_\xi(m):= \Delta\cdot \sum^{N- 1}_{n=0} x_\eta(n) e^{-2\pi i(m+ \eta)(n+\xi)/N} \] and its inverse \[ x_\eta(n)= d\cdot \sum^{N- 1}_{m= 0} X_\xi(m) e^{2\pi i(m+ \eta)(n+\xi)/N}, \] where \(\Delta\cdot d= 1/N\). By using an aliasing approach, one can show for \(X\in L_2(\mathbb{R})\) and its Fourier transform \(X_\xi\in L_2(\mathbb{R})\) under some assumptions that \[ x_\eta(n):= \sum_{k\in\mathbb{Z}} x((n+\xi+ kN)\Delta) e^{-2\pi ik\eta}, \] \[ X_\xi(m):= \sum_{l\in\mathbb{Z}} X((m+ \eta+lN)d) e^{2\pi il\xi} \] are related by the above \(\text{DFT}_{\xi\eta}\). Thus, the discretization error of the Fourier transform reads \[ \begin{multlined} \text{DFT}\{x((n+ \xi)\Delta)\}(m)- X((m+\eta)d)= \sum_{l\in\mathbb{Z}\backslash\{0\}} X((m+\eta+ lN)d) e^{2\pi il\xi}-\\ \text{DFT}\Biggl\{ \sum_{k\in\mathbb{Z}\backslash\{0\}} x((n+ \xi+kN)\Delta) e^{-2\pi ik\eta}\Biggr\}(m).\end{multlined} \] The authors examine the error as a function of \(\xi(\eta)\).