ISSN 0032-9460, Problems of Information Transmission, 2015, Vol. 51, No. 3, pp. 240–266.
Pleiades Publishing, Inc., 2015.
Original Russian Text
H. Boche, U.J. M¨onich, 2015, published in Problemy Peredachi Informatsii, 2015, Vol. 51, No. 3, pp. 41–69.
METHODS OF SIGNAL PROCESSING
Strong Divergence for System Approximations
and U. J. M¨onich
Technische Universit¨at M¨unchen, Lehrstuhl f¨ur Theoretische Informationstechnik, Germany
Massachusetts Institute of Technology, Research Laboratory of Electronics, USA
Received January 3, 2015
Abstract—In this paper we analyze approximation of stable linear time-invariant systems,
like the Hilbert transform, by sampling series for bandlimited functions in the Paley–Wiener
. It is known that there exist systems and functions such that the approximation
process is weakly divergent, i.e., divergent for certain subsequences. Here we strengthen this
result by proving strong divergence, i.e., divergence for all subsequences. Further, in case
of divergence, we give the divergence speed. We consider sampling at Nyquist rate as well as
oversampling with adaptive choice of the kernel. Finally, connections between strong divergence
and the Banach–Steinhaus theorem, which is not powerful enough to prove strong divergence,
Sampling theory studies reconstruction of a function in terms of its samples. In addition to
its mathematical signiﬁcance, sampling theory plays a fundamental role in modern signal and
information processing because it is the basis for today’s digital world .
The fundamental initial result of the theory states that the Shannon sampling series
sin(π(t − k))
can be used to reconstruct bandlimited functions f with ﬁnite L
-norm from their samples
. Since this initial result, many diﬀerent sampling theorems have been developed, and
determining the function classes for which the theorems hold and the mode of convergence now
constitute an entire area of research [2–5].
In this paper we study the convergence behavior of diﬀerent sampling series for the Paley–
Wiener space PW
consisting of absolutely integrable bandlimited functions. Analyzing sampling
series and ﬁnding sampling theorems for the Paley–Wiener space PW
has a long tradition [4,6,7].
Since Shannon’s initial result for PW
, eﬀorts have been made to extend it to larger signal
In this paper we prove strong divergence, i.e., divergence for all subsequences, for diﬀerent
sampling series, where only weak divergence, i.e., divergence for certain subsequences, was known
before, and further, we give the order of divergence. We also study the approximation of linear
The material in this paper was presented in part at the 2015 IEEE International Conference on Acoustics,
Speech, and Signal Processing (ICASSP’2015).
Partly supported by the German Research Foundation (DFG) under grant BO 1734/22-1.
Supported by the German Research Foundation (DFG) under grant MO 2572/1-1.