Lip α Approximation on Closed Sets with Lip α Extension (original) (raw)

Approximation of holomorphic functions by Taylor-Abel-Poisson means

Ukrainian Mathematical Journal, 2007

We investigate the problem of approximation of functions f holomorphic in the unit disk by means A f r ρ, () as ρ → 1-. In terms of the error of approximation by these means, a constructive characteristic of classes of holomorphic functions H p r Lip α is given. The problem of the saturation of A f r ρ, () in the Hardy space H p is solved.

A note on approximation theorems

Archivum mathematicum, 1979

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Weakly uniformly continuous holomorphic functions and the approximation property

Indagationes Mathematicae, 2001

We study the approximation property for spaces of FrChet and GLteaux holomorphic functions which are weakly uniformly continuous on bounded sets. We show when U is a balanced open subset of a Baire or barrelled metrizable locally convex space, E, that the space of holomorphic functions which are weakly uniformly continuous on U-bounded sets has the approximation property if and only if the strong dual of E, Ed, has the approximation property. We also character& the approximation property for these spaces of vector-valued holomorphic functions in terms of the tensor product of the corresponding space of scalar-valued holomorphic functions and the range space.

Compact holomorphic mappings on Banach spaces and the approximation property

Journal of Functional Analysis, 1976

1. Let E be a complex Banach space. It is well known that C(E\C), the space of continuous scalar-valued functions on E endowed with the compact-open topology, always has the approximation property, since there are continuous partitions of unity. However, for the space H{E\C) of holomorphic scalar-valued functions on E 9 the situation is more complicated. In §2 of this note, we describe this situation. Briefly, there is an exact analogy between the question of approximation by finite rank linear mappings on compact sets and the question of approximation by finite rank holomorphic mappings on compact sets.

On Constrained /^-Approximation of Complex Functions

2009

A function/analytic in any disc of radius greater than 1 is approximated in the Z,-sense over a class of polynomials which also interpolate/ on a subset of the roots of unity. The resulting solution is used to discuss Walsh-type equiconvergence. The main theorem of the paper generalizes certain results of Walsh, Rivlin and Cavaretta etal

Progress in Approximation Theory and Applicable Complex Analysis

2017

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Lipalpha harmonic approximation on closed sets

Proceedings of the American Mathematical Society, 2001

In this paper the Lipα harmonic approximation (0 < α < 1 2) on relatively closed subsets of a domain in the complex plane is characterized under the same conditions given by S. Gardiner for the uniform case. Thus, the result of P. Paramonov on Lipα harmonic polynomial approximation for compact subsets is extended to closed sets. Moreover, the problem of uniform approximation with continuous extension to the boundary for harmonic functions and similar questions in Lipα harmonic approximation are also studied.

A Leray-Schauder type theorem for approximable maps: A simple proof

Proceedings of the American Mathematical Society, 1998

We present a simple and direct proof for a Leray-Schauder type alternative for a large class of condensing or compact set-valued maps containing convex as well as nonconvex maps. The aim of this note is to extend the Leray-Schauder type nonlinear alternative presented in [BI] to a condensing upper semicontinuous approximable set-valued map F : X → E when X is a closed subset with nonempty interior of a locally convex topological vector space E. The proof presented here is even shorter and simpler than the one given in [BI]. In what follows E stands for a Hausdorff locally convex topological vector space with a fundamental basis N of convex, symmetric neighborhoods of the origin; if X, Y are nonempty subsets of E, then F : X → Y is a set-valued map with nonempty values (simply called map). The boundary, the interior, the closure, and the convex hull of a subset A in E are denoted by ∂A, int A, A, and co A respectively. Definition 1. F is said to be upper semicontinuous (u.s.c.) on X if and only if for any open subset V of Y , the set {x ∈ X : (2) F : X → Y is said to be approachable if it has a continuous (U, V )approximative selection for any (U, V ) ∈ N × N. A(X, Y ) denotes the class of such maps. We write A(X) for A(X, X). (3) F is said to be approximable if its restriction F | K to any compact subset K of X is approachable. Note that an approachable map is approximable (cf. [B]). Examples. It is well-known that if F is u.s.c. with nonempty convex values, then F is approachable provided X is paracompact and Y is convex (cf. [DG]). Obviously, F is approximable without conditions on X (see [C]).

𝐿𝑖𝑝𝛼 Harmonic Approximation on Closed Sets

Proceedings of the American Mathematical Society, 2001

In this paper the L i p α Lip\alpha harmonic approximation ( 0 > α > 1 2 0 > \alpha > \frac {1}{2} ) on relatively closed subsets of a domain in the complex plane is characterized under the same conditions given by S. Gardiner for the uniform case. Thus, the result of P. Paramonov on L i p α Lip\alpha harmonic polynomial approximation for compact subsets is extended to closed sets. Moreover, the problem of uniform approximation with continuous extension to the boundary for harmonic functions and similar questions in L i p α Lip\alpha harmonic approximation are also studied.