On the completeness of polynomials in a weighted space of entire functions
type: Item
instance of: scholarly article
title: On the completeness of polynomials in a weighted space of entire functions
zbMATH Open document ID: 0832.30003
author: I. O. Khachatryan
published in: Journal of Contemporary Mathematical Analysis. Armenian Academy of Sciences
publication date: 1995-10-18T00:00:00Z
review text: Let \(w\) be a weight function defined on \(L_\rho = \{z : |\arg z |= {\pi \over 2 \rho}, \rho \geq 1\}\) and let \({\mathcal C}_w (L_\rho)\) denote the space of all continuous functions on \(L_\rho\) satisfying the condition \(f(t)/w(t) \to 0\) as \(|t |\to \infty\), \(t \in L_\rho\). The topology in this space is given by the norm \(|f |= \sup_{t \in L_\rho} |f(t) |/w(t)\). Let \({\mathcal P}_w (L_\rho)\) be the closure in \({\mathcal C}_w (L_\rho)\) of the polynomials and let \({\mathcal C}^*_w (L_\rho)\) denote the set of entire functions of order \(\rho\) and of minimal type belonging to \({\mathcal C}_w (L_\rho)\). The main result of this paper is the following theorem: If \(w(t) = \exp (a |t |^\alpha)\), \(0 < \alpha < \rho/(2 \rho - 1)\), \(t \in L_\rho\), \(a > O\), then \({\mathcal P}_w (L_\rho) = {\mathcal C}^*_w (L_\rho)\).
Mathematics Subject Classification ID: 30B60
Mathematics Subject Classification ID: 30D15
zbMATH DE Number: 795463
reviewed by: Fritz Haslinger
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 795463