Can the Hanning Window be represented in the time domain I know the Hanning window, $w_{Hann}(n) = 1 - \cos \left(2\pi \frac{n}{N} \right)$, is typically applied to discrete Fourier transforms. $N$ being the total number of points in a transient signal that is to be transformed with the DFT. However, I'm trying to have a look at how this window (which is what I use to measure) affects the spectral line shape of the resultant Fourier transform.

For now I simply say my transient signal is a damped oscillator of the form 
$$f(t) = e^{-t/\tau} \cos(\omega_0 t).$$The Fourier transorm of which is well known to be a Lorentzian profile. However As I measure my data with a Hanning window I want to take the Fourier transform of
$$f(t) w_{Hann}(t)$$
This means I need the Hanning window in the time domain, which I assume to be simply
$$w_{Hann}(t) = 1 - \cos \left(2\pi \frac{t}{T} \right)$$ where $T$ is the averaging time for my Fourier transform.

Is this a good approach and a sounds assumption I can make with the Hanning window in a time domain rather than for a discrete index of points?
So what you mean is that you want the continuous-time Hann window instead of the discrete-time window?

$$w_{Hann}(t) = 1 - \cos \left(2\pi \frac{t}{T} \right)$$ 

is not correct, since it goes to 0 at 0.

Blackman &amp; Tukey give the continuous time window as:

$$D_2(\tau)=\frac 1 2 \left (1+\cos \frac{\pi \tau}{T_m} \right )$$

for $|\tau| &lt; T_m$ and 0 elsewhere.

Wolfram Alpha

https://en.wikipedia.org/wiki/Hann_function

https://archive.org/details/TheMeasurementOfPowerSpectra/page/n58