Bounds for orthogonal polynomials which hold on the whole interval of orthogonality are crucial to investigating mean convergence of orthogonal expansions, weighted approximation theory, and the structure of weighted spaces. This book focuses on a method of obtaining such bounds for orthogonal polynomials (and their Christoffel functions) associated with weights on \([-1,1]\). Levin and Lubinsky obtain such bounds for weights that vanish strongly at 1 and \(-1\). They also present uniform estimates of spacing of zeros of orthogonal polynomials and applications to weighted approximation theory.

Readership

Mathematicians interested in orthogonal polynomials, harmonic analysis, approximation theory, special functions, and potential theory.

Reviews

"Contains important ideas ... essential to anyone interested in the analysis of orthogonal polynomials."

-- Journal of Approximation Theory

Table of Contents

• Introduction and results
• Some ideas behind the proofs
• Technical estimates
• Estimates for the density functions \(\mu _n\)
• Majorization functions and integral equations
• The proof of Theorem 1.7
• Lower bounds for \(\lambda _n\)
• Discretisation of a potential: Theorem 1.6
• Upper bounds for \(\lambda _n\) : Theorems 1.2 and Corollary 1.3
• Zeros: Corollary 1.4
• Bounds on orthogonal polynomials: Corollary 1.5
• \(L_p\) norms of orthonormal polynomials: Theorem 1.8
• References