A transplantation theorem for Jacobi series in weighted Hardy spaces
type: Item
instance of: scholarly article
title: A transplantation theorem for Jacobi series in weighted Hardy spaces
zbMATH Open document ID: 1044.33006
DOI: 10.1016/S0001-8708(03)00143-9
author: Akihiko Miyachi
published in: Advances in Mathematics
publication date: 2004-08-06T00:00:00Z
review text: The author studies the Jacobi series over the set of functions \[ \phi_n^{(\alpha,\beta)}(t)=c_n(1-\cos t)^{\alpha/2+1/4}(1+\cos t)^{\beta/2+1/4} P_n^{(\alpha,\beta)}(\cos t), \] where \(c_n>0\) is a normalizing constant, \(\alpha,\beta>-1\) and \(P_n\) are Jacobi polynomials. The set \(\{\phi_n\}\) is known to be the orthonormal basis in \(L^2(0,\pi)\). Let \(L^2_{a,b}\) be the weighted \(L^2(w)\) with the weight function \(w(t)=t^a(\pi-t)^b\). It was proved by R. Askey and later extended by B. Muckenhoupt that the norm of any Fourier series over \(\phi_n\) is equivalent to the norm of cos-Fourier series with the same Fourier coefficients in \(L^2_{a,b}\). The author establishes the similar result in weighted Hardy spaces \(H^p_{a,b}\) with the parameters subject to certain relations.
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 42A16
zbMATH DE Number: 2081960
zbMATH Keywords: transplantation theorem
zbMATH Keywords: Jacobi series
zbMATH Keywords: weighted Hardy space
zbMATH Keywords: singular integral operators
reviewed by: Leonid Golinskii
MaRDI profile type: Publication
cites work: A transplantation theorem for Jacobi series
cites work: Hardy spaces, BMO, and boundary value problems for the Laplacian on a smooth domain in 𝐑^{𝐍}
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cites work: Q4442605
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cites work: Transplantation theorems and multiplier theorems for Jacobi series
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description: scientific article; zbMATH DE number 2081960