Pseudo-uniform Convexity in H^p and Some Extremal Problems on Sobolev Spaces

Abstract

We extend Newman and Keldysh theorems to the behavior of sequences of functions in H p ðÞ which explain geometric properties of discs in these spaces. Through Keldysh's theorem we obtain asymptotic results for extremal polynomials in Sobolev spaces.

Asymptotic properties of extremal polynomials corresponding to measures supported on analytic regions

“…Remark. One can in fact conclude that in the proof of Theorem 3.1, the functions f n converge to 1 in H q (C \ D) (see Theorem 1 in [1]). Theorem 3.1 yields the following Corollary, which says that the strong asymptotic behavior of the polynomials P n (µ, q) is in some sense independent of τ .…”

“…Proposition 5.1. If µ is a finite measure supported on [−2, −1] ∪ [1,2] and µ(A) = µ(−A) for all measurable sets A, then one does not have uniqueness of the L 1 -extremal polynomial P n (µ, 1) for every odd n.…”

A new approach to ratio asymptotics for orthogonal polynomials

“…n (e iθ ) p n (e iθ ; µ)2 dθ 2π = ∂D |Q n (z)| 2 dµ(z) → 1by hypothesis. Theorem 2.2 establishes uniform convergence on compact subsets of C \ D and we have just established convergence of norms so the result follows by Theorem 1 in[3]. Weak Asymptotic Measures.…”

On the asymptotic behavior of \(L_{p}\) extremal polynomials

“…We use the extension of Keldysh theorem (see theorem 2 pp. 430-431 of [1]). More precisely if one notes that in our case the singular part of the measure β is equal to zero and if one takes in consideration the transformation z → 1 z , we obtain the following version of theorem 2 of [1].…”