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

${ m Lip},alpha$ approximation on closed sets with mlip,alpha{ m lip},alphamlip,alpha extension

Canadian Mathematical Bulletin, 1995

Let F be a relatively closed subset of a domain G in the complex plane. Let f be a function that is the limit, in the Lip α norm on F, of functions which are holomorphic or meromorphic on G (0 < α < 1). We prove that, under the same conditions that give Lip α-approximation (0 < α < 1 ) on relatively closed subsets of G, it is possible to choose the approximating function m in such a way that f — m can be extended to a function belonging to lip

On the Best Linear Approximation of Holomorphic Functions

Journal of Mathematical Sciences, 2016

Let Ω be an open subset of the complex plane C and let E be a compact subset of Ω. The present survey is concerned with linear n-widths for the class H ∞ (Ω) in the space C(E) and some problems on the best linear approximation of classes of Hardy-Sobolev-type in L p-spaces. It is known that the partial sums of the Faber series give the classical method for approximation of functions f ∈ H ∞ (Ω) in the metric of C(E) when E is a bounded continuum with simply connected complement and Ω is a canonical neighborhood of E. Generalizations of the Faber series are defined for the case where Ω is a multiply connected domain or a disjoint union of several such domains, while E can be split into a finite number of continua. The exact values of n-widths and asymptotic formulas for the ε-entropy of classes of holomorphic functions with bounded fractional derivatives in domains of tube type are presented. Also, some results about Faber's approximations in connection with their applications in numerical analysis are mentioned.

The approximation property for spaces of holomorphic functions on infinite-dimensional spaces I

Journal of Approximation Theory, 2004

For an open subset U of a locally convex space E; let ðHðUÞ; t 0 Þ denote the vector space of all holomorphic functions on U; with the compact-open topology. If E is a separable Fre´chet space with the bounded approximation property, or if E is a (DFC)-space with the approximation property, we show that ðHðUÞ; t 0 Þ has the approximation property for every open subset U of E: These theorems extend classical results of Aron and Schottenloher. As applications of these approximation theorems we characterize the spectra of certain topological algebras of holomorphic mappings with values in a Banach algebra. r

Aspects of approximation theory for functions of one complex variable

Journal of Soviet Mathematics, 1976

In this article we examine the results in recent years on the possibility of the approximation of functions, given on subsets of a plane, by polynomials and by rational functions in uniform and integral metrics. The first part, dealing with uniform approximations, has been written by Mel'nikov; the second part, on approximations in the mean, was written by Sinanyan. The parts are independent of each other and can be read separately.

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