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A Levi-Civitá equation on compact groups and nonabelian Fourier analysis. (English) Zbl 1202.39019

Let \(G\) be a compact group. The authors derive some properties of solutions \(w,f_i,g_i\in C(G)\) of the non-classical Levi-Cività functional equation \[ w(xy)+ w(yx)= \sum^m_{i=1}f_i(x)g_i(y),\;x,y\in G. \] They solve it completely for \(m\leq 2\). In particular they show that the solutions \((w,f)\) with \(w\neq 0\) of \(w(xy)+ w(yx)= f(x)w(y)+ f(y)w(x)\), \(x,y\in G\), can be described as follows:
(i) \(w= a\chi\) and \(f=\chi/2\), where \(a\in\mathbb{C}\setminus\{0\}\) and \(\chi\in C(G)\) is a unitary character.
(ii) \(w= a(\chi-\chi')\) and \(f= (\chi+\chi')/2\), where \(a\in\mathbb{C}\setminus\{0\}\) and \(\chi,\chi'\in C(G)\) are different unitary characters.
(iii) There is a continuous unitary irreducible representation \(\pi\) of \(G\) on \(\mathbb{C}^2\) and a \(2\times 2\) complex matrix \(W\) with trace \(0\) such that \(w=\text{tr}(W\pi(\cdot))\) and \(f={1\over 2}\,\text{tr\,}\pi\).
The present paper does not assume that \(f\) is a pre-d’Alembert function in contrast to T. M. K. Davison [Publ. Math. 75, No. 1–2, 41–66 (2009; Zbl 1212.39034)].

MSC:

39B52 Functional equations for functions with more general domains and/or ranges
22C05 Compact groups
43A65 Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis)
43A77 Harmonic analysis on general compact groups

Citations:

Zbl 1212.39034
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References:

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