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EXPLICIT SOLUTION FORMULAS FOR THE MATRIX-KP
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- Journal:
- Glasgow Mathematical Journal / Volume 51 / Issue A / February 2009
- Published online by Cambridge University Press:
- 01 February 2009, pp. 147-155
- Print publication:
- February 2009
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On the Generalized d’Alembert's and Wilson's Functional Equations on a Compact Group
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- Journal:
- Canadian Mathematical Bulletin / Volume 48 / Issue 4 / 01 December 2005
- Published online by Cambridge University Press:
- 20 November 2018, pp. 505-522
- Print publication:
- 01 December 2005
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Let
$G$ be a compact group.Let
$\sigma$ be a continuous involution of $G$. In this paper, we are concerned by the following functional equation
$$\int\limits_{G}{f\left( xty{{t}^{-1}} \right)dt}\,+\int\limits_{G}{f\left( xt\sigma \left( y \right){{t}^{-1}} \right)dt}=2g\left( x \right)h\left( y \right),\,\,\,x,y\in G,$$where
$f,g,h:G\mapsto \mathbb{C}$, to be determined, are complex continuous functions on $G$ such that 
$f$ is central. This equation generalizes d’Alembert's and Wilson's functional equations. We show that the solutions are expressed by means of characters of irreducible, continuous and unitary representations of the group $G$.
