Research PaperNo Access

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

Jyoti

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

E-mail Address: jyoti.sheoran3@gmail.com

Search for more papers by this author

and

Lalit Kumar Vashisht

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

E-mail Address: lalitkvashisht@gmail.com

Corresponding author.

Search for more papers by this author

https://doi.org/10.1142/S0219691322500072Cited by:3 (Source: Crossref)

Abstract

Dual frames are generalized Riesz bases which have potential applications in signal processing. In this paper, the construction of dual frames of matrix-valued wave packet systems in the matrix-valued function space $L^{2} (ℝ^{d}, ℂ^{s \times r})$ from dual pairs of atomic wave packet frames in $L^{2} (ℝ^{d})$ is studied. A class of matrix-valued dual generators from its associated dual pair of atomic wave packets has been obtained. We provide a characterization of matrix-valued dual window functions in terms of orthogonality of wave packet Bessel sequences. A perturbation result with respect to window functions for matrix-valued dual frames is given. It is well known that an orthogonal Parseval Hilbert frame for a Hilbert space turned out be an orthonormal basis for the space, however, this is not true for an orthogonal matrix-valued wave packet Parseval frame for the underlying matrix-valued function space. We give a type of matrix-valued orthonormal basis associated with an orthonormal basis of $L^{2} (ℝ^{d})$ .

Keywords:

AMSC: 42C15, 42C30, 42C40, 43A32

References

1. A. Aldroubi, C. Cabrelli, U. Molter and S. Tang, Dynamical sampling, Appl. Comput. Harmon. Anal. 42 (2017) 378–401. Crossref, Web of Science, Google Scholar
2. A. S. Antolín and R. A. Zalik, Matrix-valued wavelets and multiresolution analysis, J. Appl. Funct. Anal. 7(1–2) (2012) 13–25. Google Scholar
3. T. Bemrose, P. G. Casazza, K. Gröchenig, M. C. Lammers and R. G. Lynch, Weaving frames, Oper. Matrices 10(4) (2016) 1093–1116. Crossref, Web of Science, Google Scholar
4. M. Y. Bhat, Necessary condition and sufficient conditions for nonuniform wavelet frames in $L^{2} (K)$ , Int. J. Wavelets Multiresolut. Inf. Process. 16(1) (2018) 24. Web of Science, Google Scholar
5. G. Bhatt, B. D. Johnson and E. Weber, Orthogonal wavelet frames and vector-valued wavelet transforms, Appl. Comput. Harmon. Anal. 23 (2007) 215–234. Crossref, Web of Science, Google Scholar
6. P. G. Casazza and G. Kutyniok, Finite Frames: Theory and Applications (Birkhäuser, 2012). Google Scholar
7. P. G. Casazza, G. Kutyniok and M. C. Lammers, Duality principles in frame theory, J. Fourier Anal. Appl. 10 (2004) 383–408. Crossref, Web of Science, Google Scholar
8. P. G. Casazza, G. Kutyniok and M. C. Lammers, Duality principles, localization of frames, and Gabor theory, Proc. SPIE Wavelets XI, Vol. 5914 (2005), pp. 389–397. Crossref, Google Scholar
9. O. Christensen, An Introduction to Frames and Riesz Bases (Birkhäuser, 2016). Google Scholar
10. O. Christensen and A. Rahimi, Frame properties of wave packet systems in $L^{2} (ℝ^{d})$ , Adv. Comput. Math. 29 (2008) 101–111. Crossref, Web of Science, Google Scholar
11. A. Cordoba and C. Fefferman, Wave packets and Fourier integral operators, Comm. Partial Differential Equations 3(11) (1978) 979–1005. Crossref, Google Scholar
12. W. Czaja, G. Kutyniok and D. Speegle, The geometry of sets of parameters of wave packets, Appl. Comput. Harmon. Anal. 20 (2006) 108–125. Crossref, Web of Science, Google Scholar
13. Deepshikha and Jyoti, A note on discrete wave packet frames in $ℂ^{N}$ , Int. J. Wavelets Multiresolut. Inf. Process. 19(4) (2021) 9. Link, Web of Science, Google Scholar
14. Deepshikha and L. K. Vashisht, A note on discrete frames of translates in $ℂ^{N}$ , TWMS J. Appl. Eng. Math. 6(1) (2016) 143–149. Web of Science, Google Scholar
15. Deepshikha and L. K. Vashisht, Extension of Weyl–Heisenberg wave packet Bessel sequences to dual frames in $L^{2} (ℝ)$ , J. Class. Anal. 8(2) (2016) 131–145. Crossref, Google Scholar
16. Deepshikha and L. K. Vashisht, Extension of Bessel sequences to dual frames in Hilbert spaces, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 79(2) (2017) 71–82. Google Scholar
17. Deepshikha and L. K. Vashisht, Necessary and sufficient conditions for discrete wavelet frames in $ℂ^{N}$ , J. Geom. Phys. 117 (2017) 134–143. Crossref, Web of Science, Google Scholar
18. Deepshikha and L. K. Vashisht, On weaving frames, Houston J. Math. 44(3) (2018) 887–915. Web of Science, Google Scholar
19. Deepshikha and L. K. Vashisht, Vector-valued (super) weaving frames, J. Geom. Phys. 134 (2018) 48–57. Crossref, Web of Science, Google Scholar
20. D.-X. Ding, Generalized continuous frames constructed by using an iterated function system, J. Geom. Phys. 61 (2011) 1045–1050. Crossref, Web of Science, Google Scholar
21. Z. Fan, A. Heinecke and Z. Shen, Duality for frames, J Fourier Anal. Appl. 22(1) (2016) 71–136. Crossref, Web of Science, Google Scholar
22. D. Han and D. R. Larson, Frames, Bases and Group Representations, Memoirs of the American Mathematical Society, Vol. 147 (American Mathematical Society, 2000). Google Scholar
23. C. Heil, A Basis Theory Primer (Birkhäuser, 2011). Crossref, Google Scholar
24. E. Hernández, D. Labate, G. Weiss and E. Wilson, Oversampling, quasi-affine frames and wave packets, Appl. Comput. Harmon. Anal. 16 (2004) 111–147. Crossref, Web of Science, Google Scholar
25. A. Iosevich, C.-K. Lai and A. Mayeli, Tight wavelet frame sets in finite vector spaces, Appl. Comput. Harmon. Anal. 46 (2019) 192–205. Crossref, Web of Science, Google Scholar
26. A. J. E. M. Janssen, Duality and biorthogonality for Weyl–Heisenberg frames, J. Fourier Anal. Appl. 4 (1995) 403–436. Google Scholar
27. A. J. E. M. Janssen, The duality condition for Weyl–Heisenberg frames, in Gabor Analysis and Algorithms: Theory and Applications, eds. H. G. Feichtinger and T. Ströhmer (Birkhäuser, 1998), pp. 33–84. Crossref, Google Scholar
28. Jyoti, Deepshikha, L. K. Vashisht and G. Verma, Sums of matrix-valued wave packet frames in $L^{2} (ℝ^{d}, ℂ^{s \times r})$ , Glas. Mat. Ser. III 53(1) (2018) 153–177. Crossref, Web of Science, Google Scholar
29. Jyoti and L. K. Vashisht, On WH-packets of matrix-valued wave packet frames in $L^{2} (ℝ^{d}, ℂ^{s \times r})$ , Int. J. Wavelets Multiresolut. Inf. Process. 16(3) (2018) 22. Link, Web of Science, Google Scholar
30. Jyoti and L. K. Vashisht, $𝒦$ -Matrix-valued wave packet frames in $L^{2} (ℝ^{d}, ℂ^{s \times r})$ , Math. Phys. Anal. Geom. 21(3) (2018) 21. Crossref, Web of Science, Google Scholar
31. Jyoti and L. K. Vashisht, On matrix-valued wave packet frames in $L^{2} (ℝ^{d}, ℂ^{s \times r})$ , Anal. Math. Phys. 10(4) (2020) 66. Crossref, Web of Science, Google Scholar
32. Jyoti, L. K. Vashisht, G. Verma and Vrinder, Matrix-valued wave packet Bessel sequences and symmetric frames in $L^{2} (ℝ^{d}, ℂ^{s \times r})$ , Poincare J. Anal. Appl. 2018(2) (2018) 77–96. Google Scholar
33. F. Krahmer, G. Kutyniok and J. Lemvig, Sparsity and spectral properties of dual frames, Linear Algebra Appl. 439(4) (2013) 982–998. Crossref, Web of Science, Google Scholar
34. G. Kutyniok and W.-Q. Lim, Dualizable shearlet frames and sparse approximation, Constr. Approx. 44(1) (2016) 53–86. Crossref, Web of Science, Google Scholar
35. G. Kutyniok, V. Paternostro and F. Philipp, The effect of perturbations of operator-valued frame sequences and fusion frames on their duals, Oper. Matrices 11(2) (2017) 301–336. Crossref, Web of Science, Google Scholar
36. D. Labate, G. Weiss and E. Wilson, An approach to the study of wave packet systems, Contemp. Math. 345 (2004) 215–235. Crossref, Google Scholar
37. L. K. Vashisht and Deepshikha, Weaving properties of generalized continuous frames generated by an iterated function system, J. Geom. Phys. 110 (2016) 282–295. Crossref, Web of Science, Google Scholar
38. A. T. Walden and A. Serroukh, Wavelet analysis of matrix-valued time-series, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 458 (2002) 157–179. Crossref, Web of Science, Google Scholar
39. X. G. Xia, Orthonormal matrix valued wavelets and matrix Karhunen–Lo‘eve expansion, in Wavelets, Multiwavelets, and Their Applications, eds. A. Aldroubi and E. B. Lin, Contemporary of Mathematics (American Mathematical Society, 1998), pp. 159–175. Crossref, Google Scholar
40. X. G. Xia and B. Suter, Vector-valued wavelets and vector filter banks, IEEE Trans. Signal Process. 44 (1996) 508–518. Crossref, Web of Science, Google Scholar
41. R. A. Zalik, On MRA Riesz wavelets, Proc. Amer. Math. Soc. 135(3) (2007) 787–793. Crossref, Web of Science, Google Scholar
42. R. A. Zalik, Orthonormal wavelet systems and multiresolution analyses, J. Appl. Funct. Anal. 5(1) (2010) 31–41. Google Scholar