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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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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| author | Stephen Bruce Sontz |
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| author_sort | Stephen Bruce Sontz |
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| description | 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. |
| format | Article |
| id | doaj-art-2902f2ad25424d8cb0a38faea1aeca8d |
| institution | Kabale University |
| issn | 1687-9120 1687-9139 |
| language | English |
| publishDate | 2011-01-01 |
| publisher | Wiley |
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| series | Advances in Mathematical Physics |
| spelling | doaj-art-2902f2ad25424d8cb0a38faea1aeca8d2025-02-03T05:51:09ZengWileyAdvances in Mathematical Physics1687-91201687-91392011-01-01201110.1155/2011/365085365085The 𝐶-Version Segal-Bargmann Transform for Finite Coxeter Groups Defined by the Restriction PrincipleStephen Bruce Sontz0Centro de Investigación en Matemáticas, A.C. (CIMAT), 36240 Guanajuato, GTO, MexicoWe 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.http://dx.doi.org/10.1155/2011/365085 |
| spellingShingle | Stephen Bruce Sontz The 𝐶-Version Segal-Bargmann Transform for Finite Coxeter Groups Defined by the Restriction Principle Advances in Mathematical Physics |
| title | The 𝐶-Version Segal-Bargmann Transform for Finite Coxeter Groups Defined by the Restriction Principle |
| title_full | The 𝐶-Version Segal-Bargmann Transform for Finite Coxeter Groups Defined by the Restriction Principle |
| title_fullStr | The 𝐶-Version Segal-Bargmann Transform for Finite Coxeter Groups Defined by the Restriction Principle |
| title_full_unstemmed | The 𝐶-Version Segal-Bargmann Transform for Finite Coxeter Groups Defined by the Restriction Principle |
| title_short | The 𝐶-Version Segal-Bargmann Transform for Finite Coxeter Groups Defined by the Restriction Principle |
| title_sort | 𝐶 version segal bargmann transform for finite coxeter groups defined by the restriction principle |
| url | http://dx.doi.org/10.1155/2011/365085 |
| work_keys_str_mv | AT stephenbrucesontz thecversionsegalbargmanntransformforfinitecoxetergroupsdefinedbytherestrictionprinciple AT stephenbrucesontz cversionsegalbargmanntransformforfinitecoxetergroupsdefinedbytherestrictionprinciple |