This work concerns the representation and approximation of functions on a sphere with applications to source localization inverse problems in geodesy and medical imaging. The thesis is structured in 6 chapters as follows. Chapter 1 presents an introduction to the geodesy and M/EEG inverse problems. The inverse pro- blem (IP) consists in recovering a density inside the ball (Earth, human brain) from partially known data on the surface. Chapter 2 gives the ma- thematical background used along the thesis. The resolution of the inverse problem (IP) involves the resolution of two steps : the transmission data pro- blem (TP) and the density recovery (DR) problem. In practice, the data are only available on some region of the sphere, as a spherical cap, like the north hemisphere of the head (M/EEG) or continent(geodesy). For this purpose, in Chapter 3, we give an ecient method to build the appropriate Slepian basis on which we express the data. This is set up by using Gauss-Legendre qua- drature. The transmission data problem (Chapter 4) consists in estimating the data (spherical harmonic expansion) over the whole sphere from noisy measurements expressed in Slepian basis. The second step, density recovery (DR) problem, is detailed in Chapter 5 where we study three density models (monopolar, dipolar and inclusions). For the resolution of (DR), we use a best quadratic rational approximation method on planar sections. We give also some properties of the density and the operator which links it to the generated potential. In Chapter 6, we study the Chapters 3, 4 ans 5 from numerical point of view. We present some numerical tests to illustrate source localization results for geodesy and M/EEG problems when we dispose of partial data on the sphere.
