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The Bargmann-Fock space of slice hyperholomorphic functions is recently introduced by Alpay, Colombo, Sabadini and Salomon. In this paper, we reconsider this space and present a direct proof of its independence of the slice. We also introduce a quaternionic analogue of the classical Segal-Bargmann transform and discuss some of its basic properties. The explicit expression of its inverse is obtained and the connection to the left one-dimensional quaternionic Fourier transform is given.
We give identifications of the $q$-deformed Segal-Bargmann transform and define the Segal-Bargmann transform on mixed $q$-Gaussian variables. We prove that, when defined on the random matrix model of Sniady for the $q$-Gaussian variable, the classica
We develop isometry and inversion formulas for the Segal--Bargmann transform on odd-dimensional hyperbolic spaces that are as parallel as possible to the dual case of odd-dimensional spheres.
Let G/K be a Riemannian symmetric space of the complex type, meaning that G is complex semisimple and K is a compact real form. Now let {Gamma} be a discrete subgroup of G that acts freely and cocompactly on G/K. We consider the Segal--Bargmann trans
We consider the Segal-Bargmann transform for a noncompact symmetric space of the complex type. We establish isometry and surjectivity theorems for the transform, in a form as parallel as possible to the results in the compact case. The isometry theor
Motivated by the two-parameter free unitary Segal-Bargmann transform in the form of conditional expectation, we derive the integral transform representation of the two-parameter free unitary Segal-Bargmann transform which coincides to the large-$N$ l