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      The \(\mu\)-deformed Segal-Bargmann transform is a Hall type transform

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          Abstract

          We present an explanation of how the \(\mu\)-deformed Segal-Bargmann spaces, that are studied in various articles of the author in collaboration with Angulo, Echevarria and Pita, can be viewed as deserving their name, that is, how they should be considered as a part of Segal-Bargmann analysis. This explanation relates the \(\mu\)-deformed Segal-Bargmann transforms to the generalized Segal-Bargmann transforms introduced by B. Hall using heat kernel analysis. All the versions of the \(\mu\)-deformed Segal-Bargmann transform can be understood as Hall type transforms. In particular, we define a \(\mu\)-deformation of Hall's "Version C" generalized Segal-Bargmann transform which is then shown to be a \(\mu\)-deformed convolution with a \(\mu\)-deformed heat kernel followed by analytic continuation. Our results are generalizations and analogues of the results of Hall.

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          Most cited references12

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          On a Hilbert space of analytic functions and an associated integral transform part I

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            Do the Equations of Motion Determine the Quantum Mechanical Commutation Relations?

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              Convolution operator and maximal function for the Dunkl transform

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                Author and article information

                Journal
                30 July 2007
                2009-07-13
                Article
                0707.4359
                f665081e-af31-4c3b-a3ac-2c54796a4c93

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

                History
                Custom metadata
                46N50; 47N50 (Primary); 46E15; 81S99 (Secondary)
                Infinite Dimensional Analysis, Quantum Probability and Related Topics, Vol. 12, Issue: 2 (June 2009) 269-289
                Minor errors corrected. Some more clarifications. Final version
                math-ph math.MP

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