  In this paper, we shall be concerned with geometric functionals and excursion
probabilities for some nonlinear transforms evaluated on Fourier components of
spherical random fields. In particular, we consider both random spherical
harmonics and their smoothed averages, which can be viewed as random wavelet
coefficients in the continuous case. For such fields, we consider smoothed
polynomial transforms; we focus on the geometry of their excursion sets, and we
study their asymptotic behaviour, in the high-frequency sense. We focus on the
analysis of Euler-Poincar\'{e} characteristics, which can be exploited to
derive extremely accurate estimates for excursion probabilities. The present
analysis is motivated by the investigation of asymmetries and anisotropies in
cosmological data. The statistics we focus on are also suitable to deal with
spherical random fields which can only be partially observed, the canonical
example being provided by the masking effect of the Milky Way on Cosmic
Microwave Background (CMB) radiation data.
