Uniform boundedness of (C, 1) means of Jacobi expansions in weighted sup norms. I (The main theorems and ideas)

Abstract

Lubinsky and Totik’s decomposition [11] of the Cesaro operators σn(α,β) of Jacobi expansions is modified to prove uniform boundedness in weighted sup norms, i.e., ‖w(a,b)σn(α,β)‖∞ ≦ C‖w(a,b)f‖∞, whenever α,β ≧ −1/2 and a, b are within the square around (α/2 + 1/4, α/2 + 1/4) having a side length of 1. This approach uses only classical results from the theory of orthogonal polynomials and various estimates for the Jacobi weights. The present paper is concerned with the main theorems and ideas, while a second paper [7] provides some necessary estimations.