Hermite Distributed Approximating Functionals as Almost-Ideal Low-Pass Filters

The two-parameter family of Hermite Distributed Approximating Functionals (HDAFs) is shown to possess all properties that are essential requirements in filter design. When properly scaled, HDAFs provide an arbitrarily sharp high-frequency cut-off while retaining their smoothness. More precisely, bounds on the Fourier transform of the HDAF integral kernel show that it converges almost-uniformly to the ideal window, and that the pass and transition bands can be tuned independently to any width while preserving Gaussian decay in both time and frequency domains. The effective length of the HDAF filter in both domains is controlled by an estimate of the Heisenberg uncertainty product. In addition, a new asymptotic relationship between the HDAF and a windowed sinc function is obtained. In all calculations, we have aimed at precise error estimates that may assist numerical implementations.