Thesis Summary:

In this thesis we study multivariate orthogonal polynomials in the d-dimensional unit ball associated with a class of reflection-invariant weights, W_α,γ(x) := (1-||x||²)^α Π_i=1^d |x_i|^{γ _i} , where α > -1 y γ = (γ ₁,...,γ _d) ∈ (-1,+∞)^d. We obtain approximation properties of the related L²_α,γ-orthogonal polynomial projector measured in what we call Dunkl—Sobolev norms, characterize a certain family of Sobolev-type orthogonal polynomials associated to the weight above, obtain connection relations of a specific bivariate basis to then develop computational tools to easily operate with polynomials expanded in terms of this basis.