Abstract
This paper mainly studies the random sampling in multi-window quasi shift-invariant subspaces \(V_X(\Phi )\) of \(L^2({\mathbb {R}}^n)\), where X is a relatively separated subset in \({\mathbb {R}}^n\). Quasi shift-invariant space presents more generality and flexibility, but it still holds the similar structure of shift-invariant spaces and spline-type spaces in some sense. Based on the analysis of a positive-semidefinite Hilbert–Schmidt localization operator, we prove that if the number of random samples taken on a bounded domain is large enough, then the sampling stability for signals in a multi-window quasi shift-invariant space \(V_X(\Phi )\) with energy concentrated on the corresponding bounded domain can hold with high probability.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data Availability
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
References
Bass, R.F., Gröchenig, K.: Random sampling of multivariate trigonometric polynomials. SIAM J. Math. Anal. 36(3), 773–795 (2005)
Bass, R.F., Gröchenig, K.: Random sampling of bandlimited functions. Israel J. Math. 177(1), 1–28 (2010)
Bass, R.F., Gröchenig, K.: Relevant sampling of bandlimited functions. Illinois J. Math. 57(1), 43–58 (2013)
Feichtinger, H.G., Molter, U., Romero, J.L.: Perturbation techniques in irregular spline-type spaces. Int. J. Wavelets Multiresolut. Inf. Process. 6(2), 249–277 (2008)
Feichtinger, H.G., Onchis, D.M.: Constructive realization of dual systems for generators of multi-window spline-type spaces. J. Comput. Appl. Math. 234(12), 3467–3479 (2010)
Führ, H., Xian, J.: Relevant sampling in finitely generated shift-invariant spaces. J. Approx. Theory 240, 1–15 (2019)
Hamm, K., Ledford, J.: On the structure and interpolation properties of quasi shift-invariant spaces. J. Funct. Anal. 274(7), 1959–1992 (2018)
Jia, R.: Bessel sequences in Sobolev spaces. Appl. Comput. Harmon. Anal. 20(2), 298–311 (2006)
Jiang, Y., Li, W.: Convolution random sampling in multiply generated shift-invariant spaces of \(L^p({\mathbb{R} }^d)\). Ann. Funct. Anal. 12, 10 (2021)
Jiang, Y., Zhao, J.: Random sampling and reconstruction of signals with finite rate of innovation. Bull. Korean Math. Soc. 59(2), 285–301 (2022)
Kumar, A., Sampath, S.: Sampling and average sampling in quasi shift-invariant spaces. Numer. Funct. Anal. Optim. 41(10), 1246–1271 (2020)
Laura, D.C., Pierluigi, V.: P-Riesz bases in quasi shift-invariant spaces, Contemporary Mathematics, Frames and Harmonic. Analysis 706, 201–213 (2018)
Li, Y., Sun, Q., Xian, J.: Random sampling and reconstruction of concentrated signals in a reproducing kernel space. Appl. Comput. Harmon. Anal. 54, 273–302 (2021)
Li, Y., Wen, J., Xian, J.: Reconstruction from convolution random sampling in local shift invariant spaces. Inverse Prob. 35(12), 125008 (2019)
Lu, Y., Xian, J.: Non-uniform random sampling and reconstruction in signal spaces with finite rate of innovation. Acta Appl. Math. 169, 247–277 (2020)
Mitzenmacher, M., Upfal, M.: Probability and Computing: Randomized Algorithms and Probabilistic Analysis. Cambridge University Press (2005)
Onchis, D.M.: Increasing the image resolution using multi-windows spline-type spaces. Signal Process. 103, 195–200 (2014)
Patel, D., Sampath, S.: Random sampling in reproducing kernel subspaces of \(L^p({\mathbb{R} }^n)\). J. Math. Anal. Appl. 491(1), 124270 (2020)
Puy, G., Tremblay, N., Gribonval, R., Vandergheynst, P.: Random sampling of bandlimited signals on graphs. Appl. Comput. Harmon. Anal. 44(2), 446–475 (2018)
Rauhut, H.: Random sampling of sparse trigonometric polynomials. Appl. Comput. Harmon. Anal. 22(1), 16–42 (2007)
Tropp, J.A.: User-friendly tail bounds for sums of random matrices. Found. Comput. Math. 12(4), 389–434 (2012)
Yang, J.: Random sampling and reconstruction in multiply generated shift-invariant spaces. Anal. Appl. 17(2), 323–347 (2019)
Yang, J., Tao, X.: Random sampling and approximation of signals with bounded derivatives, J. Inequal. Appl., (2019) 107
Yang, J., Wei, W.: Random sampling in shift-invariant spaces. J. Math. Anal. Appl. 398(1), 26–34 (2013)
Acknowledgements
The project is partially supported by the Guangxi Natural Science Foundation (Nos. 2019GXNSFFA245012, 2020GXNSFAA159076), the National Natural Science Foundation of China (No. 12261025), Guangxi Science and Technology Project (No. 2021AC06001), Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The author declares that there is no conflict of interest regarding the publication of this paper.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Jiang, Y., Zhang, H. Random Sampling in Multi-window Quasi Shift-Invariant Spaces. Results Math 78, 83 (2023). https://doi.org/10.1007/s00025-023-01865-y
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00025-023-01865-y