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A Class of Functional Equations on a Locally Compact Group
Author(s) -
Akkouchi Mohamed,
Bakali Allal,
Khalil Idriss
Publication year - 1998
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610798005961
Subject(s) - bounded function , mathematics , unimodular matrix , locally compact space , locally compact group , square integrable function , unitary representation , integrable system , pure mathematics , rank (graph theory) , group (periodic table) , space (punctuation) , unitary state , discrete mathematics , mathematical analysis , lie group , combinatorics , physics , quantum mechanics , linguistics , philosophy , political science , law
Let G be a locally compact group not necessarily unimodular. Let μ be a regular and bounded measure on G . We study, in this paper, the following integral equation, E(μ)∫ G ϕ ( x t y ) d μ ( t ) = ϕ ( x ) ϕ ( y )This equation generalizes the functional equation for spherical functions on a Gel'fand pair. We seek solutions Φ in the space of continuous and bounded functions on G . If π is a continuous unitary representation of G such that π(μ) is of rank one, then tr(π(μ)π( x )) is a solution of E(μ). (Here, tr means trace). We give some conditions under which all solutions are of that form. We show that E(μ) has (bounded and) integrable solutions if and only if G admits integrable, irreducible and continuous unitary representations. We solve completely the problem when G is compact. This paper contains also a list of results dealing with general aspects of E(μ) and properties of its solutions. We treat examples and give some applications.