A generalized Radon transform on the plane
type: Item
instance of: scholarly article
title: A generalized Radon transform on the plane
zbMATH Open document ID: 1213.44001
author: Zhongkai Li
author: Futao Song
published in: Constructive Approximation
publication date: 2011-02-14T00:00:00Z
review text: The authors define a new generalized Radon transform \(R_{\alpha,\beta}\) on the plane for functions even on each variable. Such \(R_{\alpha,\beta}\) has natural connections with the bivariate Hankel transform, the generalized biaxially symmetric potential operator \(\Delta_{\alpha\beta}\), and the Jacobi polynomial \(P^{(\beta, \alpha)}_k(t)\). The authors obtain some properties of the transform \(R_{\alpha,\beta}\), as well as its dual operator \(R^*_{\alpha,\beta}\). Particularly, the authors obtain the generalized Fuglede formula and some inverse formulas for \(R_{\alpha,\beta}\) on the function space \(L^p_{\alpha,\beta}(\mathbb{R}^2_+)\) in terms of the bivariate Hankel-Riesz potentials. Moreover, the transform \(R_{\alpha,\beta}\) is used to represent the solutions of the partial differential equations \[ \sum^m_{j=1} a_j\Delta^j_{\alpha,\beta} u= f, \] and the Cauchy problem for the generalized wave equation associated with the operator \(\Delta_{\alpha,\beta}\).
Mathematics Subject Classification ID: 44A12
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 65R32
zbMATH DE Number: 5851416
zbMATH Keywords: generalized Radon transform
zbMATH Keywords: inversion formula
zbMATH Keywords: Riesz potential
zbMATH Keywords: Hankel transform
zbMATH Keywords: Jacobi polynomial
reviewed by: Dashan Fan
MaRDI profile type: Publication
full work available at URL: s00365-010-9099-2
OpenAlex ID: W2102047819
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DOI: 10.1007/S00365-010-9099-2
description: scientific article; zbMATH DE number 5851416