Uniform approximation of holomorphic functions on bounded hartogs domains in ?2 (original) (raw)

Holomorphic Approximation on Certain Weakly Pseudoconvex Domains in Cn

Mathematics

The purpose of this paper is to study the Mergelyan approximation property in L p and C k -scales on certain weakly pseudoconvex domains of finite/infinite type in C n . At the heart of our results lies the solvability of the additive Cousin problem with bounds as well as estimates of the ∂ ¯ -equation in the corresponding topologies.

On the Mergelyan approximation property on pseudoconvex domains in ℂⁿ

Proceedings of the American Mathematical Society, 1998

Let Ω \Omega be a smoothly bounded pseudoconvex domain of finite type in C n \mathbb {C}^{n} . We prove the Mergelyan approximation property in various topologies on Ω \Omega when the estimates for ∂ ¯ \overline {\partial } -equation are known in the corresponding topologies.

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

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.

Holomorphic approximation on compact pseudoconvex complex manifolds

The Journal of Geometric Analysis, 1998

Let M be a smoothly bounded compact pseudoconvex complex manifold of finite type in the sense of D'Angelo such that the complex structure of M extends smoothly up to bM. Let m be an arbitrary nonnegative integer. Let f be a function in H(M) ∩ H m (M), where H m (M) is the Sobolev space of order m. Then f can be approximated by holomorphic functions on M in the Sobolev space H m (M). Also, we get a holomorphic approximation theorem near a boundary point of finite type.

On the approximation of entire functions over Caratheodory domains

1994

Let D be a Caratheodory domain. For 1 ≤ p ≤ ∞, let L p (D) be the class of all functions f holomorphic in D such that k f k D,p = ( 1 A D |f(z)| p dxdy) 1/p < ∞, where A is the area of D. For f ∈ L p (D), set E p n(f) = inf t∈�n k f −tk D,p ; �n consists of all polynomials of degree at most n. In this paper we study the growth of an entire function in terms of approximation error in L p -norm on D.

Uniform approximation by polynomials with integral coefficients. II

Pacific Journal of Mathematics, 1968

Let A be a discrete subring of C of rank 2. Let X be a compact subset of C with transfinite diameter not less than unity or with transfinite diameter less than unity, void interior, and connected complement. In an earlier paper we characterized the complex valued functions on X which can be uniformly approximated by elements from the ring of polynomials A[z], In this paper the same problem is studied where X is a compact subset of C with transfinite diameter d(X) less than unity and with nonvoid interior. It is also studied for certain compact subsets of C n where n is any positive integer. These subsets will have the property that every continuous function holomorphic on the interior is uniformly approximable by complex polynomials. A large class of sets of this type is shown to exist.

On Bergman completeness of non-hyperconvex domains

Eprint Arxiv Math 9909164, 1999

In the paper we study the problems of the boundary behaviour of the Bergman kernel and the Bergman completeness in some classes of bounded pseudoconvex domains, which contain also non-hyperconvex domains. Among the classes for which we prove the Bergman completeness and the convergence of the Bergman kernel to infinity while tending to the boundary are all bounded pseudonvex balanced domains, all bounded Hartogs domains with balanced fibers over regular domains and some bounded Laurent-Hartogs domains.