Approximation on the complex sphere

The aim of this thesis is to study approximation of multivariate functions on the complex sphere by spherical harmonic polynomials. Spherical harmonics arise naturally in many theoretical and practical applications. We consider different aspects of the approximation by spherical harmonic which play an important role in a wide range of topics. We study approximation on the spheres by spherical polynomials from the geometric point of view. In particular, we study and develop a generating function of Jacobi polynomials and its special cases which are of geometric nature and give a new representation for the left hand side of a well-known formulae for generating functions for Jacobi polynomials (of integer indices) in terms of associated Legendre functions. This representation arises as a consequence of the interpretation of projective spaces as quotient spaces of complex spheres. In addition, we develop new elements of harmonic analysis on the complex sphere, and use these to establish Jackson's and Kolmogorov's inequalities. We apply these results to get order sharp estimates for m-term approximation. The results obtained are a synthesis of new results on classical orthogonal polynomials geometric properties of Euclidean spaces. As another aspect of approximation, we consider interpolation by radial basis functions. In particular, we study interpolation on the spheres and its error estimate. We show that the improved error of convergence in n dimensional real sphere, given in [7], remain true in the case of the complex sphere.