ISSN 0032-9460, Problems of Information Transmission, 2017, Vol. 53, No. 2, pp. 164–182.
Pleiades Publishing, Inc., 2017.
Original Russian Text
H. Boche, U.J. M¨onich, 2017, published in Problemy Peredachi Informatsii, 2017, Vol. 53, No. 2, pp. 70–90.
METHODS OF SIGNAL PROCESSING
Spaceability for Sets of Bandlimited Input Functions
and Stable Linear Time-Invariant Systems
with Finite Time Blowup Behavior
and U. J. M¨onich
Technische Universit¨at M¨unchen, Lehrstuhl f¨ur Theoretische Informationstechnik, Munich, Germany
Received April 8, 2016; in ﬁnal form, October 12, 2016
Abstract—The approximation of linear time-invariant systems by sampling series is studied
for bandlimited input functions in the Paley–Wiener space PW
, 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.
A central problem in signal processing is the approximation of linear time-invariant (LTI) sys-
tems, like the Hilbert transform or the derivative, by sampling series. For a given bandlimited
input function f and stable LTI system T , the canonical approximation process is given by
(t − k), (1)
= T sinc denotes the response of the system T to the sinc-function. The convergence
of (1) is not guaranteed and has to be checked from case to case.
In [2–4], the convergence behavior of (1) was analyzed for functions f in the Paley–Wiener space
of bandlimited functions with absolutely integrable Fourier transform. It was shown that for
each t ∈ R there exists a stable LTI system T and a function f ∈PW
(t − k)
= ∞. (2)
That is, we have ﬁnite time blowup where the approximation error grows arbitrarily large. Using
the Banach–Steinhaus theorem, it is easy to see that for a ﬁxed system T creating divergence, the
set of functions f for which (2) holds is a residual set. Equally, for a ﬁxed function f creating
divergence, the set of systems T for which (2) holds is a residual set. However, it is not clear
Supported by the Gottfried Wilhelm Leibniz Programme of the German Research Foundation (DFG).
Parts of this work were presented at the Workshop on Harmonic Analysis, Graphs and Learning at
the Hausdorﬀ Research Institute for Mathematics, Bonn, Germany, and at the 2016 European Signal
Processing Conference .