Let D be a bounded Jordan domain with rectifiable boundary and define AJ(D), the Bers space, as the space of holomorphic functionsf, such that ffif I2D dxdy D is finite, where AD is the Poincare metric for D. It is shown that the polynomials are dense in AJ(D) for q>3/2.

/pdf/on-polynomial-approximation-in-xaihz9sqee.pdf

On polynomial approximation in _

Let D be a bounded Jordan domain. Define AJ(D), the Bers space, to be the Banach space of holomorphic functions on D, such that ffD if LAD-1 dx dy is finite, where A)D(Z) is the Poincare metric for D. It is well known that the polynomials are dense in AJ(D) for 2? q 1. Since D is bounded it follows immediately that the polynomials belong to Aq(D) in this case. The question of polynomial density in Aq(D) has been considered by various authors; Bers [3] and Knopp [7] proved that the polynomials were dense in Aq(D), for q>2, without any assumption on the mapping functions Vp and q. Later Sheingorn [11] proved that the polynomials are dense in Aq(U*), 1 1, or if D is a Smirnov domain, Presented to the Society April 20, 1973; received by the editors April 2, 1973. AMS (MOS) subject classifications (1970). Primary 30A82; Secondary 30A58, 41A10.

/pdf/on-polynomial-density-in-izkc3p92ym.pdf

On polynomial density in _

Let D be a bounded Jordan domain such that f fX2-q dx d4 oo for q > 1 Here XD(Z) is the Poincare metric for D Define AP (D), the Bers space, to be the Frechet space of holomorphic functions f on D, such that jlfljPf r f '2-'plflP dx is finite, 0 p > 1 It is well known that the polynomials are dense in AP (D) for qp > 2 We show that they are dense in AP (D) for qp > 1 irrespective whether the boundary of D is rectifiable or not

https://www.ams.org/proc/1977-062-01/S0002-9939-1977-0425139-3/S0002-9939-1977-0425139-3.pdf

Polynomial density in Bers spaces

where Z is a multiplier for G. This series, which will converge in this case if q > 1, represent an automorphic form of dimension q. (Cf. Lehner [4], Chapter III) }. The purpose of the present paper is to extend Knopp's Lemma to the case q > 1. Knopp's proof involves substantial use of the Ber's spaces of automorphic forms. Since the character of these spaces in the case q ? 2 and 1 1 at every stage of Knopp's proof, the proof turns out to be valid. However, verfication of two of the resulting statements requires techniques far outside Knopp's paper. We begin by reproducing IKnopp's proof of the Main Lemma after which

Poincare Series of Polynomials Bounded Away from Zero on a Fundamental Region

Handle everyday research tasks with reliable, citation-backed results

Your personal Research Agent to handle research tasks with citation-backed results

Popular Tasks used by Researchers

How can I help with your research?

Meet SciSpace

Get more enhanced response by uploading the PDFs you want me to reference.

No relevant PDFs in your library

SciSpace is the AI research assistant for academics. Run systematic literature reviews on 280M+ papers, and write papers with cited sources. Trusted by 1M+ students, PhDs & researchers.

SciSpace | AI for Research

Analyze PDFs

Code & Manuscripts

Funding & Grants

Literature & Patents

Medical & Clinical Data

Systematic Review

Visualize & Present

Web & Data

Build a Google Scholar-like website for your research.

Build a website

Create charts and images for your research

Create a Chart

Write a paper for submission to a journal

Draft a manuscript

Patent Search

Design eye-catching scientific posters in minutes.

Scientific Poster Generation

Systematic Literature Review

One task is running at the moment. Your messages will be shown right after.

Drag and drop or click here to browse

Loved by <highlight>1 million+</highlight> researchers

Extract a list of specific topics and their sources from unstructured text

Topics

Compare and analyze relevant papers that matches with your search

Papers

Get insights from PDFs and bookmarked papers from your library

My library

Recent searches

Try searching for:

Catch AI-generated content in scholarly and non-scholarly content

{ai} Detector

Ai Writer

Get PDF Summaries, highlighted text explanations 

Chat with PDF

Effortlessly create in-text citations and bibliographies in APA and 2,500 other formats

Citation generator

Get explanations, summaries, and answers on academic papers

Ease up your research workflow with {scispace}'s cohort of exciting AI tools

Elevate your academic writing skills and convey your ideas the way you want

Paraphraser

Explore our range of reading and writing tools

Your file is being prepared and should be ready in a few minutes. If it's a large file, it might take a bit longer. You can close this window, and we'll email you the file when it's done.

You have reached a maximum limit of <strong>{limit}</strong> columns in the table. Remove at least <strong>1</strong> column to add or create another one.

Mark Sheingorn

Author Tools

Chat about Author

Papers

Poincare Series of Polynomials Bounded Away from Zero on a Fundamental Region