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Holomorphic Hermite functions in Segal-Bargmann spaces. (English) Zbl 1423.33010

Summary: We study systems of holomorphic Hermite functions in the Segal-Bargmann spaces, which are Hilbert spaces of entire functions on the complex Euclidean space, and are determined by the Bargmann-type integral transform on the real Euclidean space. We prove that for any positive parameter which is strictly smaller than the minimum eigenvalue of the positive Hermitian matrix associated with the transform, one can find a generator of holomorphic Hermite functions whose annihilation and creation operators satisfy canonical commutation relations. In other words, we find the necessary and sufficient conditions so that some kinds of entire functions can be such generators. Moreover, we also study the complete orthogonality, the eigenvalue problems and the Rodrigues formulas.

MSC:

33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
32A37 Other spaces of holomorphic functions of several complex variables (e.g., bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA))
35S30 Fourier integral operators applied to PDEs
46E20 Hilbert spaces of continuous, differentiable or analytic functions
46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
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References:

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