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In this paper, we shall be concerned with geometric functionals and excursionprobabilities for some nonlinear transforms evaluated on Fourier components ofspherical random fields. In particular, we consider both random sphericalharmonics and their smoothed averages, which can be viewed as random waveletcoefficients in the continuous case. For such fields, we consider smoothedpolynomial transforms; we focus on the geometry of their excursion sets, and westudy their asymptotic behaviour, in the high-frequency sense. We focus on theanalysis of Euler-Poincar\'{e} characteristics, which can be exploited toderive extremely accurate estimates for excursion probabilities. The presentanalysis is motivated by the investigation of asymmetries and anisotropies incosmological data. The statistics we focus on are also suitable to deal withspherical random fields which can only be partially observed, the canonicalexample being provided by the masking effect of the Milky Way on CosmicMicrowave Background (CMB) radiation data.

High-frequency asymptotics for Lipschitz-Killing curvatures of excursion sets on the sphere