Uncertainty inequalities and order of magnitude of Hankel transforms
type: Item
instance of: scholarly article
zbMATH Open document ID: 0980.46026
author: Jorge J. Betancor
author: Lourdes Rodríguez-Mesa
publication date: 2001-11-05T00:00:00Z
Mathematics Subject Classification ID: 46F12
zbMATH DE Number: 1668782
zbMATH Keywords: Hankel transform
MaRDI profile type: Publication
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title: Uncertainty inequalities and order of magnitude of Hankel transforms
published in: Publicationes Mathematicae Debrecen
review text: The introduction of the paper gives a very comprehensive review of the classical results for the Hankel transform. The transform is defined as, NEWLINE\[NEWLINE h_\mu(f)(x)=\int^\infty_0(xy)^{-\mu}J_\mu(xy)f(y)y^{2\mu+1}dy, NEWLINE\]NEWLINE where \(f\) is a function, which is transformable. NEWLINENEWLINENEWLINEThe paper studies the behaviour of the Hankel integral transform at infinity in the Cesaro sense. Several results surround this notion and one result is as follows: NEWLINENEWLINENEWLINEProposition. Assume that \(f\in L_{1,\mu}\cap L_{p,\mu}\), with \(1<p\leq 2\), and that \(0<\alpha+\frac{2(\mu+1)}p<2\). Then NEWLINE\[NEWLINE \lim_{\lambda \to \infty}\frac 1{\lambda^2}\int^\lambda_0 \left(1-\left(\frac t\lambda \right)^2\right)^{-1/p}t^\alpha h_\mu (f)(t)t^{2\mu+1}dt=0. NEWLINE\]NEWLINE NEWLINENEWLINENEWLINEThe proofs are elegant and complete.NEWLINENEWLINENEWLINEThe paper also establishes uncertainty inequalities for the transform and Laguerre expansions. A very nice Laguerre expansion result is as follows: NEWLINENEWLINENEWLINEProposition. Let \(f\in L_{2,\mu}\cap L_{2,\mu+2}\). Then NEWLINE\[NEWLINE \int^\infty_0x^6(|f(x)|^2+|h_\mu(f)(x)|^2)x^{2\mu+1}dx=2\sum^\infty_{n=0} |\langle x^2f,\phi^\mu_n\rangle_\mu|^2(\mu+2n+1). NEWLINE\]NEWLINENEWLINENEWLINENEWLINEThe paper concludes with an entropy inequality for the Hankel transform.
reviewed by: John Schmeelk
description: scientific article; zbMATH DE number 1668782