Uniform boundedness of (C, 1) means of Jacobi expansions in weighted sup norms. II (Some necessary estimations)

Abstract

The paper is concerned with bounds for integrals of the type

$$ \int\limits_{U_k (x)} {\left| {p_n^{(\alpha , \beta )} (t)} \right|^p w^{(a, b)} } (t) dt, p \geqq 0 $$

, involving Jacobi polynomials p ^(α,β) _n and Jacobi weights w ^(a,b) depending on α,β, a, b > −1, where the subsets U _k (x) ⊂ [−1, 1] located around x and are given by $U_k (x) = \left[ {x - \tfrac{{\phi _k (x)}} {k}, x + \tfrac{{\phi _k (x)}} {k}} \right] \cap [ - 1, 1]$ with $\phi _k (x) = \sqrt {1 - x^2 } + \tfrac{1} {k}$. The functions to be integrated will also be of the type $\left| {\tfrac{{p_n^{(\alpha , \beta )} (t)}} {{x - t}}} \right|$ on the domain [−1,1] t/ U _k (x). This approach uses estimates of Jacobi polynomials modified Jacobi weights initiated by Totik and Lubinsky in [1]. Various bounds for integrals involving Jacobi weights will be derived. The results of the present paper form the basis of the proof of the uniform boundedness of (C, 1) means of Jacobi expansions in weighted sup norms in [3].