Memoirs of the American Mathematical Society 1994; 146 pp; softcover Volume: 111 ISBN10: 0821825992 ISBN13: 9780821825990 List Price: US$41 Individual Members: US$24.60 Institutional Members: US$32.80 Order Code: MEMO/111/535
 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
