A Spherical Harmonic Analysis of the Ooty Wide Field Array (OWFA)
  Visibility Signal
  Considering redshifted $21$-cm intensity mapping with the upcoming OWFA whose
field of view subtends $\sim 57^{\circ}$ in the N-S direction, we present a
formalism which relates the measured visibilities to the spherical harmonic
coefficients of the sky signal. We use this to calculate window functions which
relate the two-visibility correlations {\it i.e.} the correlation between the
visibilities measured at two baselines and two frequencies, to different
multipoles of the multi-frequency angular power spectrum
$C_{\ell}(\nu_1,\nu_2)$. The formalism here is validated using simulations. We
also present approximate closed form analytical expressions which can be used
to calculate the window functions. Comparing the widely adopted flat sky
approximation, we find that its predictions match those of our spherical
harmonic formalism to within $16 \%$ across the entire OWFA baseline range. The
match improves at large baselines where we have $< 5 \%$ deviations.
