Advanced Search
Article Contents
Article Contents

On the polyanalytic short-time Fourier transform in the quaternionic setting

  • * Corresponding author

    * Corresponding author 
Abstract Full Text(HTML) Related Papers Cited by
  • In this paper, we consider a quaternionic short-time Fourier transform (QSTFT) with normalized Hermite functions as windows. It turns out that such a transform is based on the recent theory of slice polyanalytic functions on quaternions. Indeed, we will use the notions of true and full slice polyanalytic Fock spaces and Segal-Bargmann transforms. We prove new properties of this QSTFT including a Moyal formula, a reconstruction formula and a Lieb's uncertainty principle. These results extend a recent paper of the authors which studies a QSTFT having a Gaussian function as a window.

    Mathematics Subject Classification: Primary: 44A15, 30G35, 46E22; Secondary: 42C15.


    \begin{equation} \\ \end{equation}
  • 加载中
  • [1] L. D. Abreu, On the structure of Gabor and super Gabor spaces, Monat. Math., 161 (2010), 237-253.  doi: 10.1007/s00605-009-0177-0.
    [2] L. D. Abreu, Sampling and interpolation in Bargmann-Fock spaces of polyanalytic functions, Appl. Comp. Harm Anal., 29 (2010), 287-302.  doi: 10.1016/j.acha.2009.11.004.
    [3] L. D Abreu and H. G Feichtinger, Function spaces of Polyanalytic Functions, Harmonic and complex analysis and its applications in Trends Math., Birkhäuser/Springer, Cham, 161 (2014), pp38. doi: 10.1007/978-3-319-01806-5_1.
    [4] D. Alpay, F. Colombo and I. Sabadini, Slice Hyperholomorphic Schur Analysis, Volume 256 of Operator Theory: Advances and Applications, Birkhäuser, Basel, 2017. doi: 10.1007/978-3-319-42514-6.
    [5] D. Alpay, F. Colombo and I. Sabadini, The Fock space in the slice hyperholomorphic setting, in Hypercomplex analysis: new perspectives and Applications. Trends in Mathematics (eds.S.Bernstein, (ed)), Birkhäuser, (2014), 43–59.
    [6] D. Alpay, K. Diki and I. Sabadini, On Slice polyanalytic Functions of a Quaternionic variable, Results Math., 74 (2019), 25 pp. doi: 10.1007/s00025-018-0942-2.
    [7] D. Alpay, K. Diki and I. Sabadini, Correction to: On slice polyanalytic functions of a quaternionic variable, Results Math., 76 (2021), 4 pp. doi: 10.1007/s00025-021-01364-y.
    [8] D. AlpayK. Diki and I. Sabadini, On the global operator and Fueter mapping theorem for slice polyanalytic functions, Anal. Appl. (Singap.), 19 (2021), 941-964.  doi: 10.1142/S0219530520500189.
    [9] D. Alpay, F. Colombo, K. Diki and I. Sabadini, On a polyanalytic approach to noncommutative de Branges-Rovnyak spaces and Schur analysis, Int. Equ. Operat. Theor., 93 (2021), 63 pp. doi: 10.1007/s00020-021-02649-1.
    [10] D. Alpay, F. Colombo, K. Diki and I. Sabadini, Poly slice monogenic functions, Cauchy formulas and the PS-functional calculus, arXiv: 2011.13912.
    [11] M. Bahria and R. Ashino, Two-dimensional quaternionic wiondow Fourier Transform, in Fourier Transforms - Approach to Scientific Principles (eds. G.S. Nikolic), InTechOpen, (2011), 16 pp.
    [12] M. Balk, Polyanalytic Functions, Akademie-Verlag, Berlin, 1991.
    [13] A. BenahmandiA. El Hamyani and A. Ghanmi, S-Polyregular Bargmann Spaces, Adv. Appl. Clifford Algebras, 29 (2019), 1-30.  doi: 10.1007/s00006-019-1005-9.
    [14] F. ColomboJ. O. Gonzalez-Cervantes and I. Sabadini, Further properties of the Bergman spaces of slice regular functions, Adv. Geom., 15 (2015), 469-484.  doi: 10.1515/advgeom-2015-0022.
    [15] F. Colombo and J. Gantner, Quaternionic Closed Operators, Fractional Powers and Fractional Diffusion Processes, Operator Theory: Advances and Applications, 274, Birkhäuser/Springer Cham, 2019. doi: 10.1007/978-3-030-16409-6.
    [16] F. Colombo, J. Gantner and P. Kimsey, Spectral theory on the S-spectrum for quaternionic operators, Operator Theory: Advances and Applications, 270, Birkhäuser/Springer, Cham, 2018.
    [17] F. Colombo, I. Sabadini and D. C. Struppa, Noncommutative functional calculus, Progress in Mathematics, volume 289, Birkhäuser/Springer, Basel, 2011. doi: 10.1007/978-3-0348-0110-2.
    [18] F. Colombo, I. Sabadini and D. C. Struppa, Entire Slice Regular Functions, SpringerBriefs in Mathematics, Springer, Cham, 2016. doi: 10.1007/978-3-319-49265-0.
    [19] F. Colombo, I. Sabadini and D. C. Struppa, Michele Sce's Works in Hypercomplex Analysis. A Translation with Commentaries, Birkhäuser/Springer Basel AG, Basel, 2020. doi: 10.1007/978-3-030-50216-4.
    [20] P. Cerejeiras and U. Kähler, Monogenic Signal Theory, in Operator Theory (D.Alpay), Springer, 2014.
    [21] H. De Bie, Fourier Transforms in Clifford analysis, in Operator Theory (D.Alpay), Springer, 2014. doi: 10.1002/mma.2679.
    [22] A. De Martino, On the Clifford short-time Fourier transform and its properties, Appl. Math. Comput., 418 (2022), 20 pp. doi: 10.1016/j.amc.2021.126812.
    [23] A. De Martino and K. Diki, On the quaternionic short-time Fourier and Segal-Bargamann transfroms, Mediterr. J. Math., 18 (2021), 22 pp. doi: 10.1007/s00009-021-01745-1.
    [24] K. Diki and A. Ghanmi, A quaternionic analogue for the Segal-Bargamann transfrom, Complex. Anal. Oper. Theory, 11 (2017), 457-473.  doi: 10.1007/s11785-016-0609-5.
    [25] A. El Hamyani and A. Ghanmi, On some analytic properties of slice poly-regular Hermite polynomials, Math. Methods Appl. Sci., 41 (2018), 7985-8002.  doi: 10.1002/mma.5264.
    [26] G. Gentili, C. Stoppato and D. C. Struppa, Regular Functions of a Quaternionic Varaible, Springer Monographs, Berlin, 2013. doi: 10.1007/978-3-642-33871-7.
    [27] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Seventh Edition, Accademic Press LTD, Mathematics/Engineering, 2007.
    [28] R. GhiloniW. Moretti and A. Perotti, Continous slice functional calculus in quaternionic Hilbert spaces, Rev. Math. Phys., 25 (2013), 1350006.  doi: 10.1142/S0129055X13500062.
    [29] K. Gröchening, Foundations of Time-Frequency Analysis, Birkhäuser, Boston, 2001.
    [30] A. Intissar and A. Intissar, Spectral properties of the Cauchy transform on $L_2 (\mathbb{C}, e^{-| z|^2} d\lambda (z))$, J. Math. Anal. Appl., 313 (2006), 400-418.  doi: 10.1016/j.jmaa.2005.09.056.
    [31] W. D. KirwinJ. MourãoJ. P. Nunes and T. Qian, Extending coherent state transforms to Clifford analysis, J. Math. Phys., 57 (2016), 103505.  doi: 10.1063/1.4964448.
    [32] Y. FuL. LiU. Kaehler and P. Cerejeiras, On the Fock space of metaanalytic functions, J. Math. Anal. Appl., 414 (2014), 176-187.  doi: 10.1016/j.jmaa.2013.12.058.
    [33] N. N. Lebedev, Special Functions and Their Applications, Physico-Technical Institute, Academy of Sciences, U.S.S.R. 1972.
    [34] Y. Neretin, Lectures on Gaussian Integral Operators and Classical Groups, EMS Series of Lectures in Mathematics, European Mathematical Society (EMS), Zürich, 2011. doi: 10.4171/080.
    [35] M. Niethammer, C. Eisenhardt, J. J. Laurence, J. Jarzynski and J. Qu, Application of the short time Fourier transform to interpret ultrasonic signals, Rev. Progress Quant. Nondest. Eval., (2000), 703–910.
    [36] D. Pena PenaI. Sabadini and F. Sommen, Segal-Bargmann-Fock modules of monogenic functions, J. Math Phys., 58 (2017), 103507.  doi: 10.1063/1.5008651.
    [37] T. Qian, Generalization of Fueters result to $\mathbb{R}^{n+1}$, Rend. Mat. Acc. Lincei, 9 (1997), 111-117. 
    [38] M. Rusli, Application of Short-Time Fourier Transform and Wavelet Transformfor Sound Source Localization Using Single Moving Microphone in Machine Condition Monitoring, KnE Eng., 1 (2016), 1-6. 
    [39] M. Sce, Osservazioni sulle serie di potenze nei moduli quadratici, Atti Accad. Naz. Lincei. Rend. CI. Sci.Fis. Mat. Nat., 23 (1957), 220-225. 
    [40] K. Thirulogasanthar and A. Twareque, Regular subspaces of a quaternionic Hilbert space from quaternionic Hermite polynomials and associated coherent states, Math. Phys., 54 (2013), 013506.  doi: 10.1063/1.4774963.
    [41] N. L. Vasilevski, Poly-Fock Spaces, in Differential Operators and Related Topics, vol.I (Odessa, 1997), pp.371-386, Oper. Theory Adv. Appl., vol 117, Birkäuser, (2000), 371–386.
  • 加载中

Article Metrics

HTML views(147) PDF downloads(135) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint