Spaceability for sets of bandlimited input functions and stable linear time-invariant systems with finite time blowup behavior
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The approximation of linear time-invariant systems by sampling series is studied for bandlimited input functions in the Paley–Wiener space PW π 1 , i.e., bandlimited signals with absolutely integrable Fourier transform. It has been known that there exist functions and systems such that the approximation process diverges. In this paper we identify a signal set and a system set with divergence, i.e., a finite time blowup of the Shannon sampling expression. We analyze the structure of these sets and prove that they are jointly spaceable, i.e., each of them contains an infinite-dimensional closed subspace such that for any function and system pair from these subspaces, except for the zero elements, we have divergence.
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- 1.Boche, H. and Mönich, U.J, Signal and System Spaces with Non-Convergent Sampling Representation, in Proc. 24th European Signal Processing Conf. (EUSIPCO’2016), Budapest, Hungary, Aug. 29–Sept. 2, 2016, pp. 2131–2135.Google Scholar
- 22.Boche, H. and Mönich, U.J, System Representations for the Paley–Wiener Space PW2 p, accepted for publication in J. Fourier Anal. Appl., 2017, doi:10.1007/s00041-016-9517-3.Google Scholar