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Fractional vector-valued nonuniform MRA and associated wavelet packets on \(L^2\big ({\mathbb {R}},{\mathbb {C}}^M\big )\)

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Abstract

A generalization of Mallat’s classical multiresolution analysis, based on the theory of spectral pairs, was considered in two articles by Gabardo and Nashed. In this setting, the associated translation set is no longer a discrete subgroup of \({\mathbb {R}}\) but a spectrum associated with a certain one-dimensional spectral pair and the associated dilation is an even positive integer related to the given spectral pair. In this paper, we continue the study based on this nonstandard setting and introduce fractional vector-valued nonuniform multiresolution analysis (Fr-VNUMRA) where the associated subspace of the function space has an orthonormal basis. We establish a necessary and sufficient condition for the existence of associated wavelets and derive an algorithm for the construction of fractional vector-valued nonuniform multiresolution analysis starting from a vector refinement mask with appropriate conditions. Nevertheless, to extend the scope of the present study, we worked to construct the associated wavelet packets for such an MRA and investigate their properties by means of fractional Fourier transform.

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References

  1. Bhat, M. Y., Dar, A. H.: Wavelets packets associated with linear canonical transform on spectrum. Int. J. Wavelets Multr. Info. Process. 19(6), Art. 2150030 (2021)

  2. Bhat, M.Y., Dar, A.H.: Multiresolution analysis for linear canonical \(S\) transform. Adv. Opr. Thy. 68(6), 1–11 (2021)

    Google Scholar 

  3. Bhat, M. Y., Dar, A. H.: Vector-valued nonuniform multiresolution analysis associated with linear canonical transform. arXiv:2011.08339v1 [math.FA] 2 Nov 2020

  4. Bhat, M. Y., Dar, A. H.: Convolution and correlation theorems for Wigner-Ville distribution associated with the quaternion offset linear canonical transform. Signal, Image and Video Processing (2022). DOI: https://doi.org/10.1007/s11760-021-02074-2

  5. Chen, Q., Chang, Z.: A study on compactly supported orthogonal vector-valued wavelets and wavelet packets. Chaos, Solitons Fract. 31, 1024–1034 (2007)

    Article  MathSciNet  Google Scholar 

  6. Dai, H., Zheng, Z., Wang, W.: A new fractional wavelet transform. Commun. Nonlinear Sci. Numer. Simulat. 44, 19–36 (2017)

    Article  MathSciNet  Google Scholar 

  7. Gabardo, J.P., Nashed, M.: Nonuniform multiresolution analyses and spectral pairs. J. Funct. Anal. 158, 209–241 (1998)

    Article  MathSciNet  Google Scholar 

  8. Gabardo, J. P., Nashed, M.: An analogue of Cohen’s condition for nonuniform multiresolution analyses. In: Wavelets, Multiwavelets and Their Applications, A. Aldroubi, E. Lin (Eds.), Amer. Math. Soc., Providence, RI, 41–61 (1998)

  9. Mallat, S.G.: Multiresolution approximations and wavelet orthonormal bases of \(L^2({\mathbb{R}})\). Trans. Amer. Math. Soc. 315, 69–87 (1989)

    MathSciNet  MATH  Google Scholar 

  10. Madych, W. R.: Some elementary properties of multiresolution analysis of \(L^2({\mathbb{R} }^n)\). In: Wavelets: A Tutorial in Theory and Applications (Ed. C. K. Chui), Academic Press Inc., 259–293 (1992)

  11. Mendlovic, D., Zalevsky, Z., Mas, D., Garcia, J., Ferreira, C.: Fractional wavelet transform. Appl. Opt. 36, 4801–4806 (1997)

    Article  Google Scholar 

  12. Mendlovic, D., Zalevsky, Z., Lohmann, A.W.: Signal spatial-filtering using the localized fractional Fourier transform. Opt. Commun. 126, 14–18 (1996)

    Article  Google Scholar 

  13. Namias, V.: The fractional order Fourier transform and its application to quantum mechanics. J. Inst. Math. Appl. 25, 241–264 (1980)

    Article  MathSciNet  Google Scholar 

  14. Navascues, M.A., Massopust, P.R.: Fractal convolution: a new operation between functions. Fract. Calc. Appl. Anal. 22(3), 619–643 (2019). https://doi.org/10.1515/fca-2019-0035

    Article  MathSciNet  MATH  Google Scholar 

  15. Navascues, M.A., Mohapatra, R.N., Chand, A.K.B.: Some properties of the fractal convolution of functions. Fract. Calc. Appl. Anal. 24(6), 1735–1757 (2021). https://doi.org/10.1515/fca-2021-0075

    Article  MathSciNet  MATH  Google Scholar 

  16. Ozaktas, H., Mendlovic, D.: Fourier transforms of fractional order and their optical interpretation. Opt. Commun. 101, 163–169 (1993)

    Article  Google Scholar 

  17. Ozaktas, H., Mendlovic, D.: Fractional Fourier optics. J. Opt. Soc. Amer. 12, 743–751 (1995)

    Article  Google Scholar 

  18. Shah, F. A., Bhat, M. Y.: Vector-valued nonuniform multiresolution on local fields. Int. J. Wavelets Multiresolut. Inf. Process. 13, Art. 1550029 (2015)

  19. Shah, F.A., Debnath, L.: Fractional wavelet frames in \(L^2({\mathbb{R}})\). Fract. Calc. Appl. Anal. 21(2), 399–422 (2018). https://doi.org/10.1515/fca-2018-0023

    Article  MathSciNet  MATH  Google Scholar 

  20. Srivastava, H.M., Shah, F.A., Lone, W.Z.: Fractional nonuniform multiresolution analysis in \(L^(R)\). Math. Meth Appl. Sci. 44(2), 9351–9372 (2020)

    MathSciNet  MATH  Google Scholar 

  21. Shi, J., Zhang, N.T., Liu, X.P.: A novel fractional wavelet transform and its applications. Sci. China Inf. Sci. 55, 1270–1279 (2012)

    Article  MathSciNet  Google Scholar 

  22. Shi, J., Liu, X., Zhang, N.: Multiresolution analysis and orthogonal wavelets associated with fractional wavelet transform. Sig. Imag. Video Process. 9(1), 211–220 (2015)

    Article  Google Scholar 

  23. Upadhyay, S.K., Dubey, J.K.: Wavelet convolution product involving fractional Fourier transform. Fract. Calc. Appl. Anal. 20(1), 173–189 (2017). https://doi.org/10.1515/fca-2017-0009

    Article  MathSciNet  MATH  Google Scholar 

  24. 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 

  25. Xie, P.Z., He, J.X.: Multiresolution analysis and Haar wavelets on the product of Heisenberg group. Int. J. Wavelets Multiresolut. Inf. Process. 7(2), 243–254 (2009)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This work is supported by the Research Grant (No. JKST &IC/SRE/J/357-60) provided by JKSTIC, Govt. of Jammu and Kashmir, India.

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  1. Department of Mathematical Sciences, Islamic University of Science and Technology, Awantipora, Pulwama, Jammu and Kashmir, 192122, India

    M. Younus Bhat & Aamir H. Dar

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  1. M. Younus Bhat
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  2. Aamir H. Dar
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Correspondence to M. Younus Bhat.

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Bhat, M.Y., Dar, A.H. Fractional vector-valued nonuniform MRA and associated wavelet packets on \(L^2\big ({\mathbb {R}},{\mathbb {C}}^M\big )\). Fract Calc Appl Anal 25, 687–719 (2022). https://doi.org/10.1007/s13540-022-00035-1

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