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Operational calculus and integral transforms for groups with finite propagation speed

Gordon Blower and Ian Doust ORCID logo


Let A be the generator of a strongly continuous cosine family (cos(tA))t 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(tΔ) 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.

MSC 2010: 47A60; 47D09

Funding statement: This research was partially supported by a Scheme 2 Grant from the London Mathematical Society.


Gordon Blower thanks the University of New South Wales for hospitality.


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Received: 2015-8-31
Revised: 2016-8-30
Accepted: 2017-3-28
Published Online: 2017-5-25
Published in Print: 2017-10-1

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