Spatio–Spectral Limiting on Replacements of Tori by Cubes
Abstract
:1. Introduction
2. Background
2.1. Discrete Tori and (Hyper)Cube Graphs
2.2. Replacements of Tori by Cubes
3. Laplacian Spectrum of
3.1. Augmented Laplacian
3.2. Laplacian Spectrum of ,
4. Spatio–Spectral Limiting
4.1. Radial-Type Eigenvectors of , d = 1 Case
4.2. Examples
5. Discussion
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A. Proof of Lemma 2
References
- Jain, A.K.; Ranganath, S. Extrapolation algorithms for discrete signals with application in spectral estimation. IEEE Trans. Acoust. Speech Signal Process. 1981, 29, 830–845. [Google Scholar] [CrossRef]
- Strohmer, T. On discrete band-limited signal extrapolation. In Mathematical Analysis, Wavelets, and Signal Processing (Cairo, 1994); American Mathematical Society: Providence, RI, USA, 1995; pp. 323–337. [Google Scholar]
- Zemen, T.; Mecklenbräuker, C.F. Time-variant channel estimation using discrete prolate spheroidal sequences. IEEE Trans. Signal Process. 2005, 53, 3597–3607. [Google Scholar] [CrossRef]
- Thomson, D. Spectrum estimation and harmonic analysis. Proc. IEEE 1982, 70, 1055–1096. [Google Scholar] [CrossRef]
- Hogan, J.A.; Lakey, J. Duration and Bandwidth Limiting. Prolate Functions, Sampling, and Applications; Birkhäuser: Boston, MA, USA, 2012. [Google Scholar]
- Slepian, D. Prolate spheroidal wave functions, Fourier analysis and uncertainty. IV. Extensions to many dimensions; generalized prolate spheroidal functions. Bell Syst. Tech. J. 1964, 43, 3009–3057. [Google Scholar] [CrossRef]
- Hogan, J.A.; Lakey, J. An analogue of Slepian vectors on Boolean hypercubes. J. Fourier Anal. Appl. 2019, 25, 2004–2020. [Google Scholar] [CrossRef]
- Tsitsvero, M.; Barbarossa, S.; Di Lorenzo, P. Signals on graphs: Uncertainty principle and sampling. IEEE Trans. Signal Process. 2016, 64, 4845–4860. [Google Scholar] [CrossRef]
- Van De Ville, D.; Demesmaeker, R.; Preti, M.G. When Slepian meets Fiedler: Putting a focus on the graph spectrum. IEEE Signal Process. Lett. 2017, 24, 1001–1004. [Google Scholar] [CrossRef]
- Georgiadis, K.; Adamos, D.A.; Nikolopoulos, S.; Laskaris, N.; Kompatsiaris, I. Covariation informed graph Slepians for motor imagery decoding. IEEE Trans. Neural Syst. Rehabil. Eng. 2021, 29, 340–349. [Google Scholar] [CrossRef]
- Perraudin, N.; Ricaud, B.; Shuman, D.I.; Vandergheynst, P. Global and local uncertainty principles for signals on graphs. APSIPA Trans. Signal Inf. Process. 2018, 7, E3. [Google Scholar] [CrossRef]
- Isufi, E.; Banelli, P.; Lorenzo, P.D.; Leus, G. Observing and tracking bandlimited graph processes from sampled measurements. Signal Process. 2020, 177, 107749. [Google Scholar] [CrossRef]
- Landau, H.J.; Widom, H. Eigenvalue distribution of time and frequency limiting. J. Math. Anal. Appl. 1980, 77, 469–481. [Google Scholar] [CrossRef]
- Landau, H.J. On the density of phase-space expansions. IEEE Trans. Inform. Theory 1993, 39, 1152–1156. [Google Scholar] [CrossRef]
- Slepian, D. Some comments on Fourier analysis, uncertainty and modeling. SIAM Rev. 1983, 25, 379–393. [Google Scholar] [CrossRef]
- Xu, W.Y.; Chamzas, C. On the periodic discrete prolate spheroidal sequences. SIAM J. Appl. Math. 1984, 44, 1210–1217. [Google Scholar] [CrossRef]
- Yang, X.; Wu, L.; Zhang, H. A space-time spectral order sinc-collocation method for the fourth-order nonlocal heat model arising in viscoelasticity. Appl. Math. Comput. 2023, 457, 128192. [Google Scholar] [CrossRef]
- Wang, W.; Zhang, H.; Jiang, X.; Yang, X. A high-order and efficient numerical technique for the nonlocal neutron diffusion equation representing neutron transport in a nuclear reactor. Ann. Nucl. Energy 2024, 195, 110163. [Google Scholar] [CrossRef]
- Zhang, H.; Yang, X.; Tang, Q.; Xu, D. A robust error analysis of the OSC method for a multi-term fourth-order sub-diffusion equation. Comput. Math. Appl. 2022, 109, 180–190. [Google Scholar] [CrossRef]
- Zhou, Z.; Zhang, H.; Yang, X. H1-norm error analysis of a robust ADI method on graded mesh for three-dimensional subdiffusion problems. Numer. Algor. 2023. [Google Scholar] [CrossRef]
- Chung, F.R.K. Spectral Graph Theory; CBMS Regional Conference Series in Mathematics; American Mathematical Society: Providence, RI, USA, 1997; Volume 92, p. xii+207. [Google Scholar]
- Hoory, S.; Linial, N.; Wigderson, A. Expander graphs and their applications. Bull. Am. Math. Soc. 2006, 43, 439–561. [Google Scholar] [CrossRef]
- Cui, Y.; Ou, J.; Liu, S. On 3-Restricted Edge Connectivity of Replacement Product Graphs. Axioms 2023, 12, 504. [Google Scholar] [CrossRef]
- Previte, J.P. Graph substitutions. Ergod. Theory Dynam. Syst. 1998, 18, 661–685. [Google Scholar] [CrossRef]
- Reed, M.; Simon, B. Methods of Modern Mathematical Physics. I. Functional Analysis, 2nd ed.; Academic Press Inc.: New York, NY, USA, 1980; p. xv+400. [Google Scholar]
- Stein, E.M.; Weiss, G. Introduction to Fourier Analysis on Euclidean Spaces; Princeton University Press: Princeton, NJ, USA, 1971; p. x+297. [Google Scholar]
- Donoho, D.L.; Stark, P.B. Uncertainty principles and signal recovery. SIAM J. Appl. Math. 1989, 49, 906–931. [Google Scholar] [CrossRef]
- Zhu, Z.; Karnik, S.; Davenport, M.A.; Romberg, J.; Wakin, M.B. The eigenvalue distribution of discrete periodic time-frequency limiting operators. IEEE Signal Process. Lett. 2018, 25, 95–99. [Google Scholar] [CrossRef]
- Karnik, S.; Romberg, J.; Davenport, M.A. Improved bounds for the eigenvalues of prolate spheroidal wave functions and discrete prolate spheroidal sequences. Appl. Comput. Harmon. Anal. 2021, 55, 97–128. [Google Scholar] [CrossRef]
- Boulsane, M.; Bourguiba, N.; Karoui, A. Discrete prolate spheroidal wave functions: Further spectral analysis and some related applications. J. Sci. Comput. 2020, 82, 54. [Google Scholar] [CrossRef]
- Hogan, J.A.; Lakey, J. Frame properties of shifts of prolate spheroidal wave functions. Appl. Comput. Harmon. Anal. 2015, 39, 21–32. [Google Scholar] [CrossRef]
- Pesenson, I.Z.; Pesenson, M.Z. Graph signal sampling and interpolation based on clusters and averages. J. Fourier Anal. Appl. 2021, 27, 39. [Google Scholar] [CrossRef]
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Hogan, J.A.; Lakey, J.D. Spatio–Spectral Limiting on Replacements of Tori by Cubes. Mathematics 2023, 11, 4714. https://doi.org/10.3390/math11234714
Hogan JA, Lakey JD. Spatio–Spectral Limiting on Replacements of Tori by Cubes. Mathematics. 2023; 11(23):4714. https://doi.org/10.3390/math11234714
Chicago/Turabian StyleHogan, Jeffrey A., and Joseph D. Lakey. 2023. "Spatio–Spectral Limiting on Replacements of Tori by Cubes" Mathematics 11, no. 23: 4714. https://doi.org/10.3390/math11234714
APA StyleHogan, J. A., & Lakey, J. D. (2023). Spatio–Spectral Limiting on Replacements of Tori by Cubes. Mathematics, 11(23), 4714. https://doi.org/10.3390/math11234714