Hilbert spaces of \(q\)-shifted factorial series
type: Item
instance of: scholarly article
title: Hilbert spaces of \(q\)-shifted factorial series
zbMATH Open document ID: 0786.33011
author: Gordon Macdonald
published in: Journal of Mathematical Analysis and Applications
publication date: 1994-01-23T00:00:00Z
review text: For \(0<q<1\) and \(\alpha\geq 0\), Hilbert spaces \({\mathcal H}_ q^{(\alpha)}\) of functions analytic on \(\text{Re}(z)>(1-\alpha)/2\) are constructed with the inner product on \({\mathcal H}_ q^{(\alpha)}\) given by integration with respect to a finite measure \(\mu_ q^{(\alpha)}\) supported on \(\text{Re}(z)\geq (1-\alpha)/2\) and such that the functions \(e_ n(z;q)= (q^{1-z};q)_ n/ (q;q)_ n\) for \(n=0,1,2,\dots\) are an orthonormal set in \({\mathcal H}_ q^{(\alpha)}\). It is shown that the \(q\)-Cesàro matrix \(C_ q\) (the matrix whose \((i,j)\)th entry is \(q^ i(1-q)/ (1-q^{i+1})\) if \(i\geq j\) and 0 if \(i<j\) for \(i,j=0,1,2,\dots\)) can be represented as the restriction of a multiplication operator to the coinvariant subspace of \({\mathcal H}_ q^{(\alpha)}\) spanned by \(\{e_ n(z;q)\}_{n=0}^ \infty\). The properties of \({\mathcal H}_ q^{(\alpha)}\) (including inner product and reproducing kernel formulas) and of multiplication operators on \({\mathcal H}_ q^{(\alpha)}\) are studied and norm and spectral bounds are obtained for \(C_ q\).
Mathematics Subject Classification ID: 33D99
Mathematics Subject Classification ID: 46E20
Mathematics Subject Classification ID: 46E22
zbMATH DE Number: 483999
zbMATH Keywords: \(q\)-factorial series
zbMATH Keywords: operators
zbMATH Keywords: Cesàro matrix
zbMATH Keywords: Hilbert spaces
zbMATH Keywords: reproducing kernel
MaRDI profile type: Publication
full work available at URL: jmaa.1993.1289
OpenAlex ID: W2072411026
DOI: 10.1006/JMAA.1993.1289
description: scientific article; zbMATH DE number 483999