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On matrix-valued wave packet frames in \(L^2({\mathbb {R}}^d, {\mathbb {C}}^{s\times r})\)

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Abstract

In this paper, we study matrix-valued wave packet frames for the matrix-valued function space \(L^2({\mathbb {R}}^d, {\mathbb {C}}^{s\times r})\). An interplay between matrix-valued wave packet frames and its associated atomic wave packet frames is discussed. This is inspired by examples which show that frame properties cannot be carried from matrix-valued wave packet scaling functions to its associated atomic wave packet scaling functions and vice versa. Construction of matrix-valued wave packet frames for \(L^2({\mathbb {R}}^d, {\mathbb {C}}^{s\times r})\) from corresponding atomic wave packet frames for \(L^2({\mathbb {R}}^d)\) (and conversely) are given. Some special classes of matrix-valued scaling functions are given. A characterization of tight matrix-valued wave packet frames in terms of orthogonality of Bessel sequences has been obtained. Further, we provide a characterization of superframes which can generate matrix-valued frames. Finally, a Paley-Wiener type perturbation result with respect to matrix-valued wave packet scaling functions is given.

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References

  1. Antolín, A.S., Zalik, R.A.: Matrix-valued wavelets and multiresolution analysis. J. Appl. Funct. Anal. 7(1–2), 13–25 (2012)

    MathSciNet  MATH  Google Scholar 

  2. Balan, R.: Multiplexing of signals using superframes. In: SPIE Wavelets Applications, Vol. 4119 of Signal and Image Processing VIII, 118–129 (2000)

  3. Balan, R.: Density and redundancy of the noncoherent Weyl–Heisenberg superframes. Contemp. Math. 247, 29–41 (1999)

    Article  MathSciNet  Google Scholar 

  4. Bhatt, G., Johnson, B.D., Weber, E.: Orthogonal wavelet frames and vector-valued wavelet transforms. Appl. Comput. Harmon. Anal. 23, 215–234 (2007)

    Article  MathSciNet  Google Scholar 

  5. Casazza, P.G., Kutyniok, G.: Finite Frames: Theory and Applications. Birkhäuser, Basel (2012)

    MATH  Google Scholar 

  6. Casazza, P.G., Christensen, O.: Perturbations of operators and applications to frame theory. J. Fourier Anal. Appl. 3, 543–557 (1997)

    Article  MathSciNet  Google Scholar 

  7. Christensen, O., Heil, C.: Perturbations of Banach frames and atomic decompositions. Math. Nach. 185, 33–47 (1997)

    Article  MathSciNet  Google Scholar 

  8. Christensen, O., Rahimi, A.: Frame properties of wave packet systems in \(L^2({\mathbb{R}}^d)\). Adv. Comput. Math. 29, 101–111 (2008)

    Article  MathSciNet  Google Scholar 

  9. Christensen, O.: An Introduction to Frames and Riesz Bases, 2nd edn. Birkhäuser, Basel (2016)

    MATH  Google Scholar 

  10. Christensen, O., Hasannasab, M., Lemvig, J.: Explicit constructions and properties of generalized shift-invariant systems in \(L^2({\mathbb{R}})\). Adv. Comput. Math. 43, 443–472 (2017)

    Article  MathSciNet  Google Scholar 

  11. Cordoba, A., Fefferman, C.: Wave packets and Fourier integral operators. Commun. Partial Differ. Equ. 3(11), 979–1005 (1978)

    Article  MathSciNet  Google Scholar 

  12. Czaja, W., Kutyniok, G., Speegle, D.: The geometry of sets of prameters of wave packets. Appl. Comput. Harmon. Anal. 20, 108–125 (2006)

    Article  MathSciNet  Google Scholar 

  13. Daubechies, I.: Ten Lectures on Wavelets. SIAM, Philadelphia (1992)

    Book  Google Scholar 

  14. Deepshikha, Vashisht, L.K.: A note on discrete frames of translates in \({\mathbb{C}}^N\). TWMS J. Appl. Eng. Math. 6(1), 143–149 (2016)

  15. Deepshikha, Vashisht, L.K.: Extension of Weyl-Heisenberg wave packet Bessel sequences to dual frames in \(L^2({\mathbb{R}})\). J. Class. Anal. 8(2), 131–145 (2016)

  16. Deepshikha, Vashisht, L.K.: Extension of Bessel sequences to dual frames in Hilbert spaces. Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 79(2), 71–82 (2017)

  17. Deepshikha, Vashisht, L.K.: Necessary and sufficient conditions for discrete wavelet frames in \({\mathbb{C}}^N\). J. Geom. Phys. 117, 134–143 (2017)

  18. Deepshikha, Vashisht, L.K.: Vector-valued (super) weaving frames. J. Geom. Phys. 134, 48–57 (2018)

  19. Ding, Dao-Xin: Generalized continuous frames constructed by using an iterated function system. J. Geom. Phys. 61, 1045–1050 (2011)

    Article  MathSciNet  Google Scholar 

  20. Duffin, R.J., Schaeffer, A.C.: A class of nonharmonic Fourier series. Trans. Am. Math. Soc. 72, 341–366 (1952)

    Article  MathSciNet  Google Scholar 

  21. de Gosson, M.: Hamiltonian deformations of Gabor frames: first steps. Appl. Comput. Harmon. Anal. 38(2), 196–221 (2014)

    Article  MathSciNet  Google Scholar 

  22. de Gosson, M.: The canonical group of transformations of a Weyl-Heisenberg frame; applications to Gaussian and Hermitian frames. J. Geom. Phys. 114, 375–383 (2017)

    Article  MathSciNet  Google Scholar 

  23. de Gosson, M., Gröchenig, K., Romero, J.L.: Stability of Gabor frames under small time Hamiltonian evolutions. Lett. Math. Phys. 106(6), 799–809 (2016)

    Article  MathSciNet  Google Scholar 

  24. Favier, S.J., Zalik, R.A.: On the stability of frames and Riesz bases. Appl. Comput. Harmon. Anal. 2(2), 160–173 (1995)

    Article  MathSciNet  Google Scholar 

  25. Gabor, D.: Theory of communication. J. Inst. Elect. Eng. 93, 429–457 (1946)

    Google Scholar 

  26. Gröchenig, K.: Foundations of Time-Frequency Analysis. Birkhäuser, Basel (2000)

    MATH  Google Scholar 

  27. Gröchenig, K., Lyubarskii, Y.: Gabor (super) frames with Hermite functions. Math. Ann. 345(2), 267–286 (2009)

    Article  MathSciNet  Google Scholar 

  28. Han, D., Larson, D.R.: Frames, Bases and Group Representations. Mem. Am. Math. Soc. 147, x+94 (2000)

    MathSciNet  MATH  Google Scholar 

  29. Heil, C., Walnut, D.: Continuous and discrete wavelet transforms. SIAM Rev. 31(4), 628–666 (1989)

    Article  MathSciNet  Google Scholar 

  30. Hernández, E., Labate, D., Weiss, G., Wilson, E.: Oversampling, quasi-affine frames and wave packets. Appl. Comput. Harmon. Anal. 16, 111–147 (2004)

    Article  MathSciNet  Google Scholar 

  31. Jyoti, Vashisht, L.K.: On WH-packets of matrix-malued wave packet frames in \(L^2({\mathbb{R}}^d, {\mathbb{C}}^{s\times r})\). Int. J. Wavelets Multiresolut. Inf. Process. 16(3), 1850022 (2018)

  32. Jyoti, Deepshikha, Vashisht, L.K., Verma, G.: Sums of matrix-valued wave packet frames in \(L^2({\mathbb{R}}^d, {\mathbb{C}}^{s\times r})\). Glas. Mat. Ser. III 53(1), 153–177 (2018)

  33. Jyoti, Vashisht, L.K.: \({\cal{K}}\)-Matrix-valued wave packet frames in \(L^2({\mathbb{R}}^d, {\mathbb{C}}^{s\times r})\). Math. Phys. Anal. Geom. 21(3), Art 21 (2018)

  34. Jyoti, Vashisht, L.K., Verma, G., Vrinder: Matrix-valued wave packet Bessel sequences and symmetric frames in \(L^2({\mathbb{R}}^d, {\mathbb{C}}^{s\times r})\). Poincare J. Anal. Appl. 2018(2), 77–96 (2018)

  35. Khattar, G., Vashisht, L.K.: The reconstruction property in Banach spaces generated by matrices. Adv. Pure Appl. Math. 5(3), 151–160 (2014)

    Article  MathSciNet  Google Scholar 

  36. Labate, D., Weiss, G., Wilson, E.: An approach to the study of wave packet systems. Contemp. Math. 345, 215–235 (2004)

    Article  MathSciNet  Google Scholar 

  37. von Neumann, J.: Mathematical Foundations of Quantum Mechanics. Princeton University Press, Princeton, NJ (1932/1945/1955)

  38. Vashisht, L.K., Deepshikha: Weaving properties of generalized continuous frames generated by an iterated function system. J. Geom. Phys. 110, 282–295 (2016)

  39. Walden, A.T., Serroukh, A.: Wavelet analysis of matrix-valued time-series. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 458, 157–179 (2002)

    Article  MathSciNet  Google Scholar 

  40. Xia, X.G., Suter, B.W.: Vector-valued wavelets and vector filter banks. IEEE Trans. Signal Process. 44(3), 508–518 (1996)

    Article  Google Scholar 

  41. Xia, X.G.: Orthonormal matrix valued wavelets and matrix Karhunen–Loève expansion. Contemp. Math. 216, 159–175 (1998)

    Article  Google Scholar 

  42. Zalik, R.A.: On MRA Riesz wavelets. Proc. Am. Math. Soc. 135(3), 787–793 (2007)

    Article  MathSciNet  Google Scholar 

  43. Zalik, R.A.: Orthonormal wavelet systems and multiresolution analyses. J. Appl. Funct. Anal. 5(1), 31–41 (2010)

    MathSciNet  MATH  Google Scholar 

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  1. Department of Mathematics, University of Delhi, Delhi, 110007, India

    Jyoti & Lalit Kumar Vashisht

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  2. Lalit Kumar Vashisht
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Correspondence to Lalit Kumar Vashisht.

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The research of Jyoti is supported by the Council of Scientific & Industrial Research (CSIR), India, Grant No. 09/045(1374)/2015-EMR-I.

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Jyoti, Vashisht, L.K. On matrix-valued wave packet frames in \(L^2({\mathbb {R}}^d, {\mathbb {C}}^{s\times r})\). Anal.Math.Phys. 10, 66 (2020). https://doi.org/10.1007/s13324-020-00417-9

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