AVERAGE SAMPLING IN MIXED SHIFT-INVARIANT SUBSPACES WITH GENERATORS IN A HYBRID-NORM SPACE
This paper mainly studies the average sampling and reconstruction in shift-invariant subspaces of mixed Lebesgue spaces where the generator φ of the shift-invariant subspace belongs to a hybrid-norm space of mixed form, which is weaker than the usual assumption of Wiener amalgam space and allows to control the orders p, q. First, the sampling stability for two kinds of average sampling functionals is established. Then, we give the corresponding iterative approximation projection algorithms with exponential convergence for reconstructing signals in the mixed shift-invariant subspaces from the average samples.
average sampling, shift-invariant signal, mixed Lebesgue space, hybrid-norm.
Received: April 8, 2022; Accepted: May 25, 2022; Published: June 14, 2022
How to cite this article: Haizhen Li and Yan Tang, Average sampling in mixed shift-invariant subspaces with generators in a hybrid-norm space, Far East Journal of Applied Mathematics 113 (2022), 45-66. http://dx.doi.org/10.17654/0972096022010
This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License
References:
[1] A. Aldroubi, Q. Sun and W. S. Tang, Nonuniform average sampling and reconstruction in multiply generated shift-invariant spaces, Constr. Approx. 20 (2004), 173-189.[2] A. Benedek, A. P. Calderón and R. Panzone, Convolution operators on Banach space valued functions, Proc. Natl. Acad. Sci. USA 48 (1962), 356-365.[3] A. Benedek and R. Panzone, The space Lp with mixed norm, Duke Math. J. 28(3) (1961), 301-324.[4] J. J. Benedetto and G. Zimmermann, Sampling multipliers and the Poisson summation formula, J. Fourier Anal. Appl. 3(5) (1997), 505-523.[5] G. Cleanthous, A. G. Georgiadis and M. Nielsen, Molecular decomposition of anisotropic homogeneous mixed-norm spaces with applications to the boundedness of operators, Appl. Comput. Harmon. Anal. 47(2) (2019), 447-480.[6] D. L. Fernandez, Vector-valued singular integral operators on Lp-spaces with mixed norms and applications, Pacific J. Math. 129(2) (1987), 257-275.[7] J. L. Francia, F. J. Ruiz and J. L. Torrea, Calderón-Zygmund theory for operator-valued kernels, Adv. Math. 62(1) (1986), 7-48.[8] A. G. Georgiadis, J. Johnsen and M. Nielsen, Wavelet transforms for homogeneous mixed-norm Triebel-Lizorkin spaces, Monatsh. Math. 183 (2017), 587-624.[9] L. Huang, L. Liu, D. Yang and W. Yuan, Dual spaces of anisotropic mixed-norm Hardy spaces, Proc. Amer. Math. Soc. 147 (2019), 1201-1215.[10] Y. C. Jiang and J. K. Kou, Semi-average sampling for shift-invariant signals in a mixed Lebesgue space, Numer. Funct. Anal. Optim. 41 (2020), 1045-1064.[11] R. Li, B. Liu, R. Liu and Q. Y. Zhang, Nonuniform sampling in principal shift-invariant subspaces of mixed Lebesgue spaces J. Math. Anal. Appl. 453 (2017), 928-941.[12] R. Li, B. Liu, R. Liu and Q. Y. Zhang, The Lp,q-stability of the shifts of finitely many functions in mixed Lebesgue spaces Acta Math. Sin. (Engl. Ser.) 34(6) (2018), 1001-1014.[13] Z. J. Song, B. Liu, Y. W. Pang, C. P. Hou and X. L. Li, An improved Nyquist-Shannon irregular sampling theorem from local averages, IEEE Trans. Inform. Theory 58(9) (2012), 6093-6100.[14] W. Sun and X. Zhou, Reconstruction of band limited signals from local averages, IEEE Trans. Inform. Theory 48 (2002), 2955-2963.[15] R. Torres and E. Ward, Leibniz’s rule, sampling and wavelets on mixed Lebesgue spaces, J. Fourier Anal. Appl. 21(5) (2015), 1053-1076.[16] E. Ward, New estimates in harmonic analysis for mixed Lebesgue spaces, Ph.D. Thesis, University of Kansas, 2010.[17] Q. Y. Zhang, Reconstruction of functions in principal shift-invariant subspaces of mixed Lebesgue spaces, 2018. arXiv: 1801. 05715v2.[18] Q. Y. Zhang, Nonuniform average sampling in multiply generated shift-invariant sub-spaces of mixed Lebesgue spaces, Int. J. Wavelets Multiresolut. Inf. Process. 18(3) (2020), 2050013.
