We study linear extremal problems in the Bergman space A p of the unit disc for p an even integer. Given a functional on the dual space of A p with representing kernel k ∈ A q , where 1/p + 1/q = 1, we show that if the Taylor coefficients of k are sufficiently small, then the extremal function F ∈ H ∞ . We also show that if q ≤ q 1 < ∞, then F ∈ H (p−1)q 1 if and only if k ∈ H q 1 . These results extend and provide a partial converse to a theorem of Ryabykh.
Abstract.In this study we analyze 53 magnetic clouds (MCs) of standard profiles observed in WIND magnetic field and plasma data, in order to estimate the speed of MC expansion (V E ) at 1 AU, where the expansion is investigated only for the component perpendicular to the MCs' axes. A high percentage, 83%, of the good and acceptable quality cases of MCs (N(good)=64) were actually expanding, where "good quality" as used here refers to those MCs that had relatively well determined axial attitudes. Two different estimation methods are employed. The "scalar" method (where the estimation is denoted V E,S ) depends on the average speed of the MC from Sun-to-Earth (
Abstract. In this paper, we study the problem of finding the extremal element for a linear functional over a uniformly convex Banach space. We show that a unique extremal element exists and depends continuously on the linear functional, and vice versa. Using this, we simplify and clarify Ryabykh's proof that for any linear functional on a uniformly convex Bergman space with kernel in a certain Hardy space, the extremal function belongs to the corresponding Hardy space.Our purpose is to study extremal problems over uniformly convex Banach spaces. For a given linear functional φ, what elements x of the space with norm x = 1 maximize Re φ(x)? Because of the uniform convexity, this problem will always have a unique solution, which is called the extremal element. In this paper, we show that the extremal element depends continuously on the functional φ, and it can be approximated by the solutions of the same extremal problem over subspaces of the original Banach space.Moreover, we show that an extremal element can arise from at most one linear functional of unit norm, and that the functional depends continuously on the extremal element. Using these results, we give a streamlined proof of a theorem of Ryabykh, which says that for a functional defined on the Bergman space A p , with kernel in the Hardy space H q , the extremal element is in H p , where 1 < p < ∞ and 1 p + 1 q = 1. Uniform convexity and extremal problemsLet X be a complex Banach space and let X * be its dual space. For a given linear functional φ ∈ X * with φ = 0, we are interested in all elements x ∈ X with norm x = 1 such that (1.1)Re φ(x) = sup y =1Re φ(y) = φ .Such a problem is referred to as an extremal problem, and x is an extremal element.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
customersupport@researchsolutions.com
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.