A Sharp Error Estimate for Numerical Fourier Transform of Band-Limited Functions Based on Windowed Samples

W. Dickmeis and R. J. Nessel published their first version of a quantitative extension of the classical uniform boundedness principle in [J. Approx. Theory 31 (1981), 161-174]. It is a general approach to finding counterexamples that prove sharpness of error estimates. So far applications of this principle include error bounds for approximation processes, cubature rules, ordinary and partial differential equations, and reconstruction from samples. Here we discuss the error of discrete approximations of the Fourier transform based on windowed samples for band-limited functions. The results can be applied to the Hann- and Blackmann-Harris-window but also to window-functions that enable higher orders of convergence. We describe a class of such windows.