The weighted Lp-norms of orthonormal polynomials for Erdös weights

✍ Scribed by D.S. Lubinsky

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 552 KB
• Volume: 33
• Category: Article
• ISSN: 0898-1221
• DOI: 10.1016/s0898-1221(96)00227-1

✦ Synopsis

R ---* R is even, and "smooth," and of faster than polynomial growth at infinity. For example, we consider Q(x) = exp k/(]xla), a > 1, where expk = exp(exp(.., exp(... ))) denotes the k th iterated exponential. Weights of the form W 2 for such W are often called ErdSs weights. We compute the growth of the Lp-norms (0 < p < oo) of the weighted orthonormal polynomials Pn (W 2, x)W(x) for a large class of Erd6s weights, based on recent work of the author with Levin and Mthembu on the Loo-norm of pn(W 2, x)W(x). As an auxiliary result, we obtain bounds on the fundamental polynomials of Lagrange interpolation at the zeros of pn(W2,x), and as a corollary, we deduce finer spacing for the zeros of pn (W 2, .). The growth of the Lp-norms of orthonormal polynomials is a key factor in investigating convergence of orthogonal expansions and Lagrange interpolation in Lp-norms.