TY - JOUR
T1 - Strong divergence for system approximations
AU - Boche, H.
AU - Mönich, U. J.
N1 - Publisher Copyright:
© 2015, Pleiades Publishing, Inc.
PY - 2015/7/1
Y1 - 2015/7/1
N2 - 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 space PWπ 1. 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, are discussed.
AB - 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 space PWπ 1. 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, are discussed.
UR - http://www.scopus.com/inward/record.url?scp=84944447229&partnerID=8YFLogxK
U2 - 10.1134/S0032946015030047
DO - 10.1134/S0032946015030047
M3 - Article
AN - SCOPUS:84944447229
SN - 0032-9460
VL - 51
SP - 240
EP - 266
JO - Problems of Information Transmission
JF - Problems of Information Transmission
IS - 3
ER -