A unified construction of product formulas and convolutions for Sturm-Liouville operators
type: Item
instance of: scholarly article
title: A unified construction of product formulas and convolutions for Sturm-Liouville operators
zbMATH Open document ID: 1478.34029
author: Rúben Sousa
author: Manuel Guerra
author: S. Yakubovich
published in: Analysis and Mathematical Physics
publication date: 2021-06-07T00:00:00Z
review text: This is a voluminous article on unification of product formulas and convolutions for Sturm-Liouville operators. The abtsract of the article reflects the content in the best way. ``We establish a positive product formula for the solutions of the Sturm-Liouville equation \(l(u) = \lambda u\), where \(l\) belongs to a general class which includes singular and degenerate Sturm-Liouville operators. Our technique relies on a positivity theorem for possibly degenerate hyperbolic Cauchy problems and on a regularization method which makes use of the properties of the diffusion semigroup generated by the Sturm-Liouville operator. We show that the product formula gives rise to a probability preserving convolution algebra structure on the space of finite measures which satisfies the basic axioms for developing harmonic analysis on the convolution algebra. Unlike previous works, our framework includes a subfamily of Sturm-Liouville operators for which the support of the convolution of Dirac measures is noncompact. The connection with hypergroup theory is discussed. Convolution-type integral equations on weighted Lebesgue spaces are also discussed, and a solvability condition is established.''
reviewed by: Erdoǧan Şen
Mathematics Subject Classification ID: 34B24
Mathematics Subject Classification ID: 35L80
Mathematics Subject Classification ID: 43A10
Mathematics Subject Classification ID: 43A62
zbMATH DE Number: 7355863
zbMATH Keywords: hypergroup
zbMATH Keywords: Sturm-Liouville operator
zbMATH Keywords: product formula
zbMATH Keywords: generalized convolution
zbMATH Keywords: degenerate hyperbolic equation
zbMATH Keywords: convolution integral equations
MaRDI profile type: Publication
full work available at URL: s13324-021-00520-5
OpenAlex ID: W3151853761
Wikidata QID: Q113899305
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DOI: 10.1007/S13324-021-00520-5