Abstract : This thesis focuses on the use of second generation wavelet frames for the statistical study of spherical random fields. Among phenomena modeled by such a field, we are specifically interested in the Cosmic Microwave Background (CMB).
The localization of needlets (recent construction of wavelets) on the sphere is considered and optimized in terms of spatial concentration and statistical estimation. These functions are then used to define a new estimator of the angular power spectrum. The study of this estimator's theoretical (in the high-frequency asyptotics) and practical properties shows that it improves existing methods in a realistic model including missing data and heteroscedastic noise. Beside spectral estimation, the use of needlets is also introduced in a component separation framework.
Four introductory chapters are devoted to physical, analytical and statistical aspects of the CMB, and to the presentation of our results. Then, we present four articles (in collaboration): "Practical wavelet design on the sphere" ; "CMB power spectrum estimation using wavelets" ; "Spectral estimation on the sphere with needlets: high frequency asymptotics" and "A full sky, low foreground, high resolution CMB map from WMAP".