Document Type : Research Paper

Authors

1 Lab. P.D.E., Algebra and Spectral Geometry, Department of mathematics, Faculty of sciences, P.O.Box 133, Ibn Tofail University in Kenitra; Morocco.

2 Analysis, P.D.G $&$ Spectral Geometry. Lab. M.I.A.-S.I., CeReMAR, Department of Mathematics, P.O. Box 1014, Faculty of Sciences, Mohammed V University in Rabat, Morocco.

3 Faculty of Law, Economics and Social Sciences, Ibn Zohr University, Agadir, Morocco.

10.22130/scma.2021.140216.875

Abstract

We define in a natural way the bicomplex analog of the frames (bc-frames) in the setting of  bicomplex infinite Hilbert spaces, and we characterize them in terms of their idempotent components. We also extend some classical results from frames theory to bc-frames and show that some of them do not remain valid for bc-frames in general. The construction of bc-frame operators and Weyl--Heisenberg bc-frames are also discussed.

Keywords

[1] D.S. Alexiadis and G.D. Sergiadis, Estimation of motions in color image sequences using hypercomplex Fourier transforms, IEEE Trans. Image Process., 18 (1) (2009), pp. 168-187.

[2] P. Balazs, N. Holighaus, T. Necciari and D. Stoeva, Frame theory for signal processing in psychoacoustics, in: Excursions in harmonic analysis, vol.5, Appl. Numer. Harmon. Anal., Birkhauser, Springer, Cham, 2017, pp. 225-268.

[3] J.J. Benedetto, Irregular sampling and frames in Wavelets, in: A Tutorial in Theory and Applications, Chui. C.K. ed. Cambridge, MA. Academic Press, 1992, pp. 445-507.

[4] H. Bolcskei, F. Hlawatsch and H.G. Feichtinger, Frame-theoretic analysis of oversampled filter banks, IEEE Trans. Signal Process., 46 (12) (1998), pp. 3256-3268.

[5] P.G. Casazza, The art of frame theory, Taiwanese J. Math., 4 (2) (2000), pp. 129-202.

[6] P.G. Casazza and G. Kutyniok, Frames of subspaces, Contemp. Math., 345 (2004), pp. 87-113.
 
[7] P.G. Casazza and R.G. Lynch, A brief introduction to Hilbert space frame theory and its applications, Finite frame theory, Proc. Sympos. Appl. Math., 73 (2016), pp. 1-51.

[8] O. Christensen, An Introduction to Frames and Riesz Bases, Birkhauser, 2016.

[9] I. Daubechies, Ten Lectures on Wavelets, Philadelphia. SIAM, 1992.

[10] I. Daubechies, The wavelet transformation, time-frequency localization and signal analysis, IEEE Trans. Inform. Theory., 36 (1990), pp. 961-1005.

[11] I. Daubechies, A. Grassmann and Y. Meyer, Painless nonorthogonal expansions, J. Math. Physics., 27 (1986), pp. 1271-1283.

[12] B. Deng, W. Schempp, C. Xiao and Z. Wu, On the existence of Weyl-Heisenberg and affine frames, Preprint, 1997.

[13] D.L. Donoho and M. Elad, Optimally sparse representation in general (nonorthogonal) dictionaries via $l^1$ minimization, in: Proc. Natl. Acad. Sci. USA, 100 (5) (2003), pp. 2197-2202.

[14] R. Duffin and A. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc., 72 (1952), pp. 341-366.

[15] Y.C. Eldar and G.D. Jr Forney, Optimal tight frames and quantum measurement, IEEE Trans. Inf. Theory., 48 (3) (2002), pp. 599-610.

[16] A. El Gourari, A. Ghanmi and K. Zine, On bicomplex Fourier-Wigner transforms, Int. J. Wavelets Multiresolut. Inf. Process., 18 (3) (2020), 2050008, 16 pages.

[17] M.T. El-Melegy and A.T. Kamal, Color Image Processing Using Reduced Biquaternions With Application to Face Recognition in a PCA Framework, Proc. IEEE Inte. Conf. Computer Vision (ICCV)., (2017), pp. 3039-3046.

[18] T.A. Ell and S.J. Sangwine, Hypercomplex Fourier transforms of color images, IEEE Trans. Image Process., 16 (1) (2007), pp. 22-35.

[19] M. Frank and D.R. Larson, Frames in Hilbert $C^*$--modules and $C^*$-algebras, preprint, University of Houston and Texas A$&$M University, Texas, U.S.A., 1998.

[20] M. Frank and D.R. Larson, A module frame concept for Hilbert $C^*$-modules, in: the Functional and Harmonic Analysis of Wavelets and frames, Contemp. Math., 247, (1999), pp. 207-233.

[21] M. Frank and D.R. Larson, Modular frames for Hilbert $C^*$-modules and symmetric approximation of frames, Proc. SPIE., 4119 (2000), pp. 325-336.

[22] M. Frank and D.R. Larson, Frames in Hilbert $C^*$-modules and $C^* $-algebras, J. Operator Theory., 2 (48) (2002), pp. 273-314.

[23] P. Gavruta, On some identities and inequalities for frames in Hilbert spaces, J. Math. Anal. Appl., 321 (2006), pp. 469-478.

[24] R. Gervais Lavoie, L. Marchildon and D. Rochon, Infinite dimensional bicomplex Hilbert spaces, Ann. Funct. Anal., 2 (2010), pp. 75-91.

[25] A. Ghanmi and K. Zine, Bicomplex analogs of Segal-Bargmann and fractional Fourier transforms, Adv. Appl. Clifford Algebr., 29 (4) (2019), 20 pages.

[26] K. Grochenig, Foundations of Time-Frequency Analysis, Birkhauser, Boston 2001.

[27] E. Guariglia and S. Silvestrov, Fractional-wavelet analysis of positive definite distributions and wavelets on $D'(mathbb{C})$, in: Engineering Mathematics II, Springer Proc. Math. Stat., 179, Springer, Cham, 2016, pp. 337-353.
 
[28] C. Heil and D. Walnut, Continuous and discrete wavelet transforms, SIAM Rev., 31 (1989), pp. 628-666 .

[29] C. Heil, A Basis Theory Primer, Birkhauser, Boston, 2010.

[30] M. Joita, Tensor products of Hilbert modules over locally $C^*$-algebras, Czechoslovak Math. J., 54 (129) (2004), pp. 727-737.

[31] A. Khosravi and B. Khosravi, Frames and bases in tensor products of Hilbert spaces and Hilbert $C^*$-modules, Proc. Indian Acad. Sci. Math. Sci., 117 (1) (2007), pp. 1-12.

[32] A. Khosravi and B. Khosravi, Fusion frames and $g$-frames in Hilbert $C^*$-modules, Int. J. Wavelet. Multiresolut. Inf. Process., 6 (3) (2008), pp. 433-446.

[33] J. Kovacevic and A. Chebira, An introduction to frames, Found. Trends Signal Process., 2 (1) (2008), pp. 1-94.

[34] C.E. Moxey, S.J. Sangwine and T. Ell, A. Hypercomplex correlation techniques for vector images, IEEE Trans. Signal Process., 51 (7) (2003), pp. 1941-1953.

[35] G.B. Price, An Introduction to Multicomplex Spaces and Functions, Monographs and Textbooks, Pure and Appl Math., 140, Marcel Dekker Inc., New York, 1991.

[36] I. Raeburn and S.J. Thompson, Countably generated Hilbert modules, the Kasparov stabilisation theorem, and frames with Hilbert modules, Proc. Amer. Math. Soc., 131 (5) (2003), pp. 1557-1564.
 
[37] D. Rochon and M. Shapiro, On algebraic properties of bicomplex and hyperbolic numbers, An Univ. Oradea Fasc. Mat., 11 (2004), pp. 71-110.

[38] D. Rochon and S. Tremmblay, Bicomplex quantum mechanics, I: The generalized Schrodinger equation, Adv. Appl. Clifford Algebr., 14 (2004), pp. 231-248.

[39] D. Rochon and S. Tremmblay, Bicomplex quantum mechanics, II: The Hilbert space, Adv. Appl. Clifford Algebr., 16 (2006), pp. 135-157.

[40] J. Wu, Frames in Hilbert $C^*$-modules, Electronic Theses and Dissertations, 899, pp. 2004-2019.

[41] R. Young, An Introduction to Nonharmonic Fourier Series, Academic Press, New York, 1980.

[42] M. Zaied, O. Jemai and C. Ben Amar, Training of the Beta wavelet networks by the frames theory: Application to face recognition, in: first Workshops on Image Processing Theory, Tools and Applications, 2008, pp. 1-6.