Hardy spaces for the strip
type: Item
instance of: scholarly article
title: Hardy spaces for the strip
zbMATH Open document ID: 1160.30021
author: Andrew Bakan
author: Sten Kaijser
published in: Journal of Mathematical Analysis and Applications
publication date: 2007-06-26T00:00:00Z
review text: For computational convenience, the authors study Hardy spaces on the strip \(\mathfrak{S} = \{ z \in \mathbb{C} \,: | \text{Im}\, z | < 1\}\). They consider three candidates to be genuine Hardy spaces on such strip. Firstly, since \(\phi(z) = \tanh \,\frac{\pi}{4}\) is a conformal maps from \(\mathfrak{S}\) onto the unit disc \(\mathbb{D}\), it is natural to define \(H^p(\mathfrak{S}) = \{ f \in \text{Hol}(\mathbb{D}) \,:\, f \circ \phi^{-1} \in H^p({\mathbb{D}})\}\), \(0< p < \infty\), and \(\| f| _{H^p(\mathfrak{S})} = \| f \circ \phi^{-1}\| _{H^p({\mathbb{D}})}\). A second class of \(H^p\)-spaces are the translation invariant spaces defined as \[ H^p_{\tau} (\mathfrak{S}) = \left\{f \in \text{Hol} (\mathfrak{S})\,: \, \| f\| ^p_{H^p_{\tau}(\mathfrak{S})} = \sup_{0 \leq y < 1} \int_{-\infty}^\infty \frac{| f(-iy + t)| ^p + | f(iy + t)| ^p}{2}\,dt < \infty \right\}, \] which sometimes are studied in the context of Paley-Wiener theory. A third class, defined as \[ H^p_{\mathcal{W}}(\mathfrak{S}) = \left\{f \in \text{Hol} (\mathfrak{S})\,: \, \| f\| ^p_{H^p_{\mathcal{W}}(\mathfrak{S})} = \sup_{0 < y < 1} \int_{-\infty}^\infty \frac{| f(-iy + t)| ^p + | f(iy + t)| ^p}{4(\cosh \frac{\pi}{2}t + \cos\frac{\pi}{2}y ) }\,dt < \infty \right\}, \] is related to the Poisson measure for the origin. The main results of the paper state that for \(0 < p < \infty\), the spaces \(H^p_{\mathcal{W}}(\mathfrak{S})\) and \(H^p_{\tau} (\mathfrak{S})\) are isometrically isomorphic (Theorem 2.1), that the spaces \(H^p_{\mathcal{W}}(\mathfrak{S})\) and \(H^p(\mathfrak{S})\) are identical with identical norms (Theorem 2.2), that functions in \(H^p_{\tau} (\mathfrak{S})\) satisfy a \(L^p\)-form of the Hadamard three lines theorem (Theorem 2.3), and that algebraic polynomials are dense in \(H^p (\mathfrak{S})\) (Theorem 2.4). It is said that an orthogonal basis of polynomials for \(H^2(\mathfrak{S})\) is presented. However, this interesting matter is just sketched in the last section. Reviewe…
Mathematics Subject Classification ID: 30H10
zbMATH DE Number: 5167521
zbMATH Keywords: \(H^{p}\)-classes
zbMATH Keywords: polynomial approximation
zbMATH Keywords: orthogonal polynomials
reviewed by: Fernando Pérez-González
MaRDI profile type: Publication
full work available at URL: j.jmaa.2006.10.088
OpenAlex ID: W2014572419
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DOI: 10.1016/J.JMAA.2006.10.088
description: scientific article; zbMATH DE number 5167521