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On matrix-valued Gabor frames over locally compact abelian groups

Uttam Kumar Sinha

Department of Mathematics, Shivaji College, University of Delhi, Delhi 110027, India

E-mail Address: [email protected]

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Lalit Kumar Vashisht

https://orcid.org/0000-0002-8634-6920

Department of Mathematics, University of Delhi, Delhi 110027, India

E-mail Address: [email protected]

Corresponding author.

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, and

Pankaj Kumar Das

Department of Mathematical Sciences, Tejpur University, Napaam, Sonitpur, Assam 784028, India

E-mail Address: [email protected]

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https://doi.org/10.1142/S0219025723500236Cited by:1 (Source: Crossref)

Abstract

In this paper, we study Gabor frames in the matrix-valued signal space $L^{2} (G, ℂ^{n \times n})$ , where $G$ is a locally compact abelian group which is metrizable and $σ$ -compact, and $n$ is a positive integer. First, we give sufficient conditions on scalars in an infinite combination of vectors (from a given matrix-valued Gabor frame) to constitute a new frame for the space $L^{2} (G, ℂ^{n \times n})$ . This generalizes a result due to Aldroubi. Second, we discuss frame conditions for finite sums of matrix-valued Gabor frames. Sufficient conditions for finite sums of matrix-valued Gabor frames in terms of frame bounds are established. It is shown that the sum of images of matrix-valued Gabor frames under bounded linear operators acting on $L^{2} (G, ℂ^{n \times n})$ constitute a frame for the space $L^{2} (G, ℂ^{n \times n})$ provided operators are adjointable with respect to the matrix-valued inner product and satisfy a majorization. Finally, we show that matrix-valued Gabor frames are stable under small perturbations.

Communicated by Uwe Franz

Keywords:

AMSC: 42C15, 42C30, 42C40

References

1. O. Christensen , An Introduction to Frames and Riesz Bases, 2nd edn. (Birkhäuser, New York, 2016). Crossref, Google Scholar
2. K. Gröchenig , Foundations of Time-Frequency Analysis (Birkhäuser, Boston, 2001). Crossref, Google Scholar
3. C. Heil , A Basis Theory Primer, Expanded edn. (Birkhäuser, New York, 2011). Crossref, Google Scholar
4. A. J. E. M. Janssen , Duality and biorthogonality for Weyl–Heisenberg frames, J. Fourier Anal. Appl. 1(4) (1995) 403–436. Crossref, Google Scholar
5. L. K. Vashisht and H. K. Malhotra , Discrete vector-valued nonuniform Gabor frames, Bull. Sci. Math. 178 (2022) 103145. Crossref, Web of Science, Google Scholar
6. R. M. Young , An Introduction to Nonharmonic Fourier Series (Academic Press, New York, 1980). Google Scholar
7. R. J. Duffin and A. C. Schaeffer , A class of nonharmonic Fourier series, Trans. Amer. Math. Soc. 72 (1952) 341–366. Crossref, Web of Science, Google Scholar
8. D. Gabor , Theory of communication, J. Inst. Elect. Eng. 93 (1946) 429–457. Google Scholar
9. I. Daubechies, A. Grossmann and Y. Meyer , Painless nonorthogonal expansions, J. Math. Phys. 27 (1986) 1271–1283. Crossref, Web of Science, Google Scholar
10. A. Aldroubi , Portraits of frames, Proc. Amer. Math. Soc. 123(6) (1995) 1661–1668. Crossref, Web of Science, Google Scholar
11. 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
12. D. Jindal, U. K. Sinha and G. Verma , Multivariate Gabor frames for operators in matrix-valued signal spaces over locally compact abelian groups, Int. J. Wavelets Multiresolut. Inf. Process. 19(2) (2021) 1–24. Link, Web of Science, Google Scholar
13. A. Krivoshein, V. Protasov and M. Skopina , Multivariate Wavelet Frames (Springer, Singapore, 2016). Google Scholar
14. I. Ya. Novikov, V. Yu. Protasov and M. A. Skopina , Wavelet Theory (Amer. Math. Soc., 2011). Crossref, Google Scholar
15. L. K. Vashisht , Banach frames generated by compact operators associated with a boundary value problem, TWMS J. Pure Appl. Math. 6(2) (2015) 254–258. Google Scholar
16. 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
17. 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
18. X. G. Xia and B. W. Suter , Vector-valued wavelets and vector filter banks, IEEE Trans. Signal Process. 44(3) (1996) 508–518. Crossref, Web of Science, Google Scholar
19. Jyoti and L. K. Vashisht , On WH-packets of matrix-malued wave packet frames in $L^{2} (ℝ^{d}, ℂ^{s \times r})$ , Int. J. Wavelets Multiresolut. Inf. Process. 16(3) (2018) 1850022. Link, Web of Science, Google Scholar
20. 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
21. 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) 1–21. Web of Science, Google Scholar
22. 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
23. Jyoti, L. K. Vashisht and U. K. Sinha , Matrix-valued frames over LCA groups for operators, Filomat 37(28), (2023) 9543–9559. Web of Science, Google Scholar
24. D. Jindal and L. K. Vashisht , Matrix-valued nonstationary frames associated with the Weyl–Heisenberg Group and extended affine group, Int. J. Wavelets Multiresolut. Inf. Process. (2023), https://doi.org/10.1142/S0219691323500224. Link, Web of Science, Google Scholar
25. D. Jindal and L. K. Vashisht, Sums of frames from the Weyl–Heisenberg group and applications to frame algorithm, arXiv:2306.09493. Google Scholar
26. H. Heuser , Functional Analysis (John Wiley, New York, 1982). Google Scholar
27. G. B. Folland , A Course in Abstract Harmonic Analysis, 2nd edn. (CRC Press, 2015). Google Scholar
28. S. J. Favier and R. A. Zalik , On the stability of frames and Riesz bases, Appl. Comput. Harmon. Anal. 2(2) (1995) 160–173. Crossref, Web of Science, Google Scholar