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Bicomplex Analogs of Segal–Bargmann and Fractional Fourier Transforms

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Abstract

We consider and discuss some basic properties of the bicomplex analog of the classical Bargmann space. The explicit expression of the integral operator connecting the complex and bicomplex Bargmann spaces is also given. The corresponding bicomplex Segal–Bargmann transform is introduced and studied as well. Its explicit expression as well as the one of its inverse are then used to introduce a class of two-parameter bicomplex Fourier transforms (bicomplex fractional Fourier transform). This approach is convenient in exploring some useful properties of this bicomplex fractional Fourier transform.

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Acknowledgements

A part of this work is done during the visit of the first author to Dipartimento di Matematica Politecnico di Milano (May–June 2017). The assistance of the members of “Ahmed Intissar’s seminar on Analysis, Partial Differential Equations and Spectral Geometry” is gratefully acknowledged.

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  1. A.-P.D.E.-G.S., CeReMAR, Department of Mathematics, Faculty of Sciences, Mohammed V University in Rabat, P.O. Box 1014 , Rabat, Morocco

    Allal Ghanmi & Khalil Zine

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  1. Allal Ghanmi
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  2. Khalil Zine
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Correspondence to Allal Ghanmi.

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Communicated by Frank Sommen.

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Ghanmi, A., Zine, K. Bicomplex Analogs of Segal–Bargmann and Fractional Fourier Transforms. Adv. Appl. Clifford Algebras 29, 74 (2019). https://doi.org/10.1007/s00006-019-0993-9

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