Abstract:
For $\alpha>0$, the Bargmann projection $P_\alpha$ is the orthogonal projection from $L^2(\gamma_\alpha)$ onto the holomorphic subspace $L^2_{hol}(\gamma_\alpha)$, where $\gamma_\alpha$ is the standard Gaussian probability measure on $\C^n$ with variance $(2\alpha)^{-n}$. The space $L^2_{hol}(\gamma_\alpha)$ is classically known as the Segal-Bargmann space. We show that $P_\alpha$ extends to a bounded operator on $L^p(\gamma_{\alpha p/2})$, and calculate the exact norm of this scaled $L^p$ Bargmann projection. We use this to show that the dual space of the $L^p$-Segal-Bargmann space $L^p_{hol}(\gamma_{\alpha p/2})$ is an $L^{p'}$ Segal-Bargmann space, but with the Gaussian measure scaled differently: $(L^p_{hol}(\gamma_{\alpha p/2}))^* \cong L^{p'}_{hol}(\gamma_{\alpha p'/2})$ (this was shown originally by Janson, Peetre, and Rochberg). We show that the Bargmann projection controls this dual isomorphism, and gives a dimension-independent estimate on one of the two constants of equivalence of the norms.

Abstract:
We consider the generalized Segal-Bargmann transform, defined in terms of the heat operator, for a noncompact symmetric space of the complex type. For radial functions, we show that the Segal-Bargmann transform is a unitary map onto a certain L^2 space of meromorphic functions. For general functions, we give an inversion formula for the Segal-Bargmann transform, involving integration against an "unwrapped" version of the heat kernel for the dual compact symmetric space. Both results involve delicate cancellations of singularities.

Abstract:
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.

Abstract:
We consider a ${\mu}$-deformation of the Segal-Bargmann transform, which is a unitary map from a ${\mu}$-deformed quantum configuration space onto a ${\mu}$-deformed quantum phase space (the ${\mu}$-deformed Segal-Bargmann space). Both of these Hilbert spaces have canonical orthonormal bases. We obtain explicit formulas for the Shannon entropy of some of the elements of these bases. We also consider two reverse log-Sobolev inequalities in the ${\mu}$-deformed Segal-Bargmann space, which have been proved in a previous work, and show that a certain known coefficient in them is the best possible.

Abstract:
We review here two reverse inequalities in the Segal-Bargmann space: a reverse hypercontractivity estimate due to Carlen and a reverse log-Sobolev inequality due to the second author.

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.

Abstract:
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 transform, defined in terms of the heat equation, on the compact quotient {\Gamma}\G/K. We obtain isometry and inversion formulas precisely parallel to the results we obtained previously for globally symmetric spaces of the complex type. Our results are as parallel as possible to the results one has in the dual compact case. Since there is no known Gutzmer formula in this setting, our proofs make use of double coset integrals and a holomorphic change of variable.

Abstract:
We present various relations among Versions A, B and C of the Segal-Bargmann transform. We get results for the Segal-Bargmann transform associated to a Coxeter group acting on a finite dimensional Euclidean space. Then analogous results are shown for the Segal-Bargmann transform of a connected, compact Lie group for all except one of the identities established in the Coxeter case. A counterexample is given to show that the remaining identity from the Coxeter case does not have an analogous identity for the Lie group case. A major result is that in both contexts the Segal-Bargmann transform for Version C is determined by that for Version A.

Abstract:
We state several equivalent noncommutative versions of the Cauchy-Riemann equations and characterize the unbounded operators on L^2(R) which satisfy them. These operators arise from the creation operator via a functional calculus involving a class of entire functions, identified by Newman and Shapiro [D. J. Newman and H. S. Shapiro, Fischer spaces of entire functions, in Entire Functions and Related Parts of Analysis (J. Koorevaar, ed.), AMS Proc. Symp. Pure Math. XI (1968), 360-369], which act as unbounded multiplication operators on Bargmann-Segal space.