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arXiv · 1509.00133

Operational calculus and integral transforms for groups with finite propagation speed

Gordon Blower·Ian Doust
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Abstract

Let $A$ be the generator of a strongly continuous cosine family $(\cos (tA))_{t\in {\bf R}}$ 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 $\cos( t\sqrt {\Delta})$ 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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Gordon Blower, Ian Doust. 2015-09-01. Operational calculus and integral transforms for groups with finite propagation speed. https://doi.org/10.1515/apam-2015-0049

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