On some Hilbert cosine functions. (English) Zbl 1157.39014
The properties and characterizations of generalized cosine functions are presented. These functions appear as solutions of equation
\[ \int_{K} \Phi(xk \cdot y) \,dk= \Phi(x) \Phi(y), \quad x,y \in G \]
(\(G\) a locally compact group, \(K\) a compact subgroup of \(\operatorname{Aut}(G)\), \(dk\) the normalized Haar measure) in the class of bounded linear operators on Hilbert space.
\[ \int_{K} \Phi(xk \cdot y) \,dk= \Phi(x) \Phi(y), \quad x,y \in G \]
(\(G\) a locally compact group, \(K\) a compact subgroup of \(\operatorname{Aut}(G)\), \(dk\) the normalized Haar measure) in the class of bounded linear operators on Hilbert space.
Reviewer: Tomasz Zgraja (Bielsko-Biała)
MSC:
| 39B32 | Functional equations for complex functions |
| 43A90 | Harmonic analysis and spherical functions |
| 39B52 | Functional equations for functions with more general domains and/or ranges |
