The 𝐶-Version Segal-Bargmann Transform for Finite Coxeter Groups Defined by the Restriction Principle
We apply a special case, the restriction principle (for which we give a definition simpler than the usual one), of a basic result in functional analysis (the polar decomposition of an operator) in order to define 𝐶𝜇,𝑡, the 𝐶-version of the Segal-Bargmann transform, associated with a finite Coxeter g...
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| Main Author: | |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2011-01-01
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| Series: | Advances in Mathematical Physics |
| Online Access: | http://dx.doi.org/10.1155/2011/365085 |
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| Summary: | We apply a special case, the restriction principle (for which we give a definition simpler than the usual one), of a basic result in functional analysis (the polar decomposition of an operator) in order to define 𝐶𝜇,𝑡, the 𝐶-version of the Segal-Bargmann transform, associated with a finite Coxeter group acting in ℝ𝑁 and a given value 𝑡>0 of Planck's constant, where 𝜇 is a multiplicity function on the roots defining the Coxeter group. Then we immediately prove that 𝐶𝜇,𝑡 is a unitary isomorphism. To accomplish this we identify the reproducing kernel function of the appropriate Hilbert space of holomorphic functions. As a consequence we prove that the Segal-Bargmann transforms for Versions 𝐴, 𝐵, and 𝐷 are also unitary isomorphisms though not by a direct application of the restriction principle. The point is that the 𝐶-version is the only version where a restriction principle, in our definition of this method, applies directly. This reinforces the idea that the 𝐶-version is the most fundamental, most natural version of the Segal-Bargmann transform. |
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| ISSN: | 1687-9120 1687-9139 |