Research PaperNo Access

A note on discrete wave packet frames in $ℂ^{N}$

Department of Mathematics, Shyampur Siddheswari Mahavidyalaya, University of Calcutta, Kolkata, West Bengal 711312, India

E-mail Address: dpmmehra@gmail.com

Search for more papers by this author

and

Jyoti

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

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

Search for more papers by this author

https://doi.org/10.1142/S0219691321500065Cited by:2 (Source: Crossref)

Abstract

We show that for every nonzero vector $ϕ$ in $ℂ^{N}$ , the discrete wave packet system ${D_{a} M_{l} T_{k} ϕ}_{a \in U (N), l, k \in 𝕀_{N}}$ constitutes a frame for the unitary space $ℂ^{N}$ . An application of this result is given, where frame conditions cannot be derived from discrete wavelet systems in $ℂ^{ℕ}$ . The canonical dual of discrete wave packet frame is also discussed.

Keywords:

AMSC: 42C15, 42C30, 42C40

References

1. P. G. Casazza and G. Kutyniok , Finite Frames: Theory and Applications (Birkhäuser, Boston, 2012). Google Scholar
2. O. Christensen , An Introduction to Frames and Riesz Bases, 2nd edn. (Birkhäuser, Boston, 2016). Crossref, Google Scholar
3. 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
4. C. Chui and X. Shi , Inequalities of LittlewoodPaley type for frames and wavelets, SIAM J. Math. Anal. 24 (1993) 263–277. Crossref, Web of Science, Google Scholar
5. A. Cordoba and C. Fefferman , Wave packets and Fourier integral operators, Comm. Partial Differential Equations 3(11) (1978) 979–1005. Crossref, Google Scholar
6. W. Czaja, G. Kutyniok and D. Speegle , The Geometry of sets of prameters of wave packets, Appl. Comput. Harmon. Anal. 20 (2006) 108–125. Crossref, Web of Science, Google Scholar
7. 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
8. 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
9. 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
10. 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
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. M. de Gosson , Hamiltonian deformations of Gabor frames: First steps, Appl. Comput. Harmon. Anal. 38(2) (2014) 196–221. Crossref, Web of Science, Google Scholar
13. D. Han, K. Kornelson, D. Larson and E. Weber , Frames for Undergraduates, Student Mathematical Library, Vol. 40 (AMS, Providence, RI, 2007). Crossref, Google Scholar
14. E. Hernández, D. Labate and G. Weiss , A unified characterization of reproducing systems generated by a finite family II, J. Geom. Anal. 12(4) (2002) 615–662. Crossref, Google Scholar
15. 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
16. A. Iosevich, C.-K. Lai and A. Mayeli , Tight wavelet frame sets in finite vector spaces, Appl. Comput. Harmon. Anal. 46(1) (2019) 192–205. Crossref, Web of Science, Google Scholar
17. 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
18. 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), Article ID: 21, 19 pp. Web of Science, Google Scholar
19. 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), Article ID: 66, 24 pp. Crossref, Web of Science, Google Scholar
20. 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), Article ID: 1850022, 22 pp. Link, Web of Science, Google Scholar
21. 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(II)) (Special Issue IWWFA-III, Delhi). Google Scholar
22. 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
23. 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