Spaceability for sets of bandlimited input functions and stable linear time-invariant systems with finite time blowup behavior
The approximation of linear time-invariant systems by sampling series is studied for bandlimited input functions in the Paley–Wiener space PWπ1, 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