I read a paper reference at http://arxiv.org/pdf/1101.1764.pdf that if we average a set $V=\{V(t_0,\nu_0), V({t_1,\nu_1),..., V(t_n,\nu_n)}\}$; with $V(t_i,\nu_i)=e^{i\sigma(t_i,\nu_i)}$ then we can approximate this average as:
$<V>\simeq sinc(\frac{\Delta \phi}{2})sinc(\frac{\Delta \theta }{2})V(t_c,\nu_c)$
$\\$where $c=(n+1)/2$ is the mid point,
$\Delta \phi=arg V(t_n,\nu_c)-argV(t_0,\nu_c)$,
$\Delta \theta=arg V(t_c,\nu_n)-argV(t_c,\nu_0)$. where arg denotes the complex argument or complex angle.
Please can some one help me to demonstrate this?
Also, if I weight this set with a window $W(t,\nu)$, what will be this approximation?
From what I understood, averaging meant weighting the data with a normalize top-hat ("box") window, and the Fourier transform of a top-hat window is a sinc. But I am not so sure that my terminology works.