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.
HDAF integral kernel is defined by Hoffman and other  as
In addition, we derive an asymptotic approximation formula for the HDAF integral kernel that shows its relation to a windowed sinc function, also referred to as Gaussian sinc-DAF by Hoffman and others (see  or ).
n](k;[sigma]) and for the HDAF integral kernel itself.
We recover the integral representation of the HDAF integral kernel by the inverse Fourier transform of (6),
The upper and lower bounds on the Fourier transform of the HDAF integral kernel are illustrated in Fig.
The scaled HDAF integral kernel of order n [member of] N is for any x [member of] R bounded by
The uncertainty product of the HDAF integral kernel [D.
Prom the functional form of the HDAF integral kernel given in Eq.
In addition, we have obtained an asymptotic relation between the HDAF integral kernel and a windowed sinc function that has proved highly useful in numerical applications and is known as the Gaussian-sinc-DAF, a sinc function windowed by a Gaussian envelope.
Instead of applying a simple high-frequency cut-off to an analog signal, we convolve it with an HDAF integral kernel and thus effectively truncate it in the frequency domain.