Discrete Harmonic Analysis related to classical orthogonal polynomials stars

Résumé

The present dissertation belongs to the so-called non-trigonometric discrete Harmonic Analysis, specifically to the one associated with classical orthogonal polynomials. Its aim is the study of the discrete analogues of some classical operators in Harmonic Analysis. To be specific, the convergence problem of the multiplier of an interval for discrete Fourier series and the problem of the norm boundedness of the transplantation operator are studied. Regarding the first problem, the multiplier of an interval related to Jacobi polynomials is defined and sufficient conditions are given to ensure its norm boundedness with weights. If we consider no weights, a characterization is provided. Moreover, the characterization of the convergence is also given. Regarding the second problem, a transplantation theorem related to Jacobi coefficients is given when we consider weighted spaces. We prove that the transplantation operators are bounded in norm with weights by means of a semi-local Calderón- Zygmund theory which has been recently furnished. Moreover, some weighted weak estimates are provided. On its behalf, a transplantation theorem for Laguerre coefficients in weighted spaces is also given. In that case, we use a discrete local Calderón- Zygmund theory which is developed in the dissertation.