Asymptotic uncorrelation for Mexican needlets

We recall Mexican needlets from [D. Geller, A. Mayeli, Continuous wavelets on compact manifolds, Math. Z. 262 (4) (2009) 895–927; D. Geller, A. Mayeli, Nearly tight frames and space-frequency analysis on compact manifolds, Math. Z. 263 (2) (2009) 234–264]. We derive an estimate for certain types of Legendre series, which we apply to the statistical properties of Mexican needlets. More precisely, we shall show that, under isotropy assumption, the Mexican needlet coefficients of a random field on the sphere are asymptotically uncorrelated, as the frequency parameter goes to infinity. This property is important in the analysis of spherical random fields, in particular in connection to the analysis of Cosmic Microwave Background (CMB) radiation data.