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Operational calculus and integral transforms for groups with finite propagation speed

Authors
  • Blower, Gordon
  • Doust, Ian
Type
Published Article
Journal
Advances in Pure and Applied Mathematics
Publisher
De Gruyter
Publication Date
May 25, 2017
Volume
8
Issue
4
Pages
265–283
Identifiers
DOI: 10.1515/apam-2015-0049
Source
De Gruyter
Keywords
License
Yellow

Abstract

Let A be the generator of a strongly continuous cosine family ( cos ⁡ ( t ⁢ A ) ) t ∈ ℝ {(\cos(tA))_{t\in\mathbb{R}}} on a complex Banach space E. The paper develops an operational calculus for integral transforms and functions of A using the generalized harmonic analysis associated to certain hypergroups. It is shown that characters of hypergroups which have Laplace representations give rise to bounded operators on E. Examples include the Mellin transform and the Mehler–Fock transform. The paper uses functional calculus for the cosine family cos ⁡ ( t ⁢ Δ ) {\cos(t\sqrt{\Delta})} which is associated with waves that travel at unit speed. The main results include an operational calculus theorem for Sturm–Liouville hypergroups with Laplace representation as well as analogues to the Kunze–Stein phenomenon in the hypergroup convolution setting.

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