Relative convolutions. I: Properties and applications. (English) Zbl 0933.43004
The author introduces the notion of relative convolutions and studies the operators of relative convolutions induced by a Lie algebra. He describes some basic properties of such convolutions (a formula for the composition of relative convolutions, etc.) and gives various applications. It is established that an algebra of relative convolutions induced by a Lie algebra \(g\) is a representation of the algebra of group convolutions on the Lie group \(\exp g\). The theory of relative convolutions is tightly connected with the theory of PDE and Heisenberg groups. The connection of this notion with some well-known classes of operators (operators of multiplication, two-sided convolutions, group convolutions, etc.) is considered. Some applications to the theory of pseudo-differential operators, to complex and hypercomplex analysis, coherent states, wavelets and quantum mechanics are also given. Contents. 1. Introduction. 2. Relative convolutions. 3. Basic properties. 4. Applications to complex and hypercomplex analysis. 5. Coherent states. 6. Applications to physics and signal theory. 7. Conclusion.
Reviewer: N.K.Karapetyants (Rostov-na-Donu)
MSC:
| 43A80 | Analysis on other specific Lie groups |
| 45P05 | Integral operators |
| 22E60 | Lie algebras of Lie groups |
| 32M05 | Complex Lie groups, group actions on complex spaces |
| 81R30 | Coherent states |
| 81S99 | General quantum mechanics and problems of quantization |
| 22E45 | Representations of Lie and linear algebraic groups over real fields: analytic methods |
Keywords:
Lie groups and algebras; relative convolutions; Heisenberg group; Clifford analysis; quantization; pseudo-differential operators; wavelets; quantum mechanics; signal theory; coherent statesReferences:
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