TY - GEN
T1 - A Banach space property for signal spaces with applications for sampling and system approximation
AU - Boche, Holger
AU - Mönich, Ullrich J.
N1 - Publisher Copyright:
© 2017 IEEE.
PY - 2017/9/1
Y1 - 2017/9/1
N2 - The Banach-Steinhaus theorem, one of the fundamental results in functional analysis, completely characterizes the convergence of linear approximation processes. If the condition of boundedness is violated, then the principle of uniform boundedness implies the unbounded divergence of the approximation process on a residual set. In this paper we give a sufficient condition for Banach spaces that guarantees the unbounded divergence not only for a residual set but rather for a set that contains an infinite dimensional closed subspace after the zero element has been added. We further show that many important signal space, e.g., Paley-Wiener and Bernstein spaces, possess this property, and demonstrate consequences for the convergence behavior of sampling series and system approximation processes.
AB - The Banach-Steinhaus theorem, one of the fundamental results in functional analysis, completely characterizes the convergence of linear approximation processes. If the condition of boundedness is violated, then the principle of uniform boundedness implies the unbounded divergence of the approximation process on a residual set. In this paper we give a sufficient condition for Banach spaces that guarantees the unbounded divergence not only for a residual set but rather for a set that contains an infinite dimensional closed subspace after the zero element has been added. We further show that many important signal space, e.g., Paley-Wiener and Bernstein spaces, possess this property, and demonstrate consequences for the convergence behavior of sampling series and system approximation processes.
UR - https://www.scopus.com/pages/publications/85031667054
U2 - 10.1109/SAMPTA.2017.8024395
DO - 10.1109/SAMPTA.2017.8024395
M3 - Conference contribution
AN - SCOPUS:85031667054
T3 - 2017 12th International Conference on Sampling Theory and Applications, SampTA 2017
SP - 484
EP - 488
BT - 2017 12th International Conference on Sampling Theory and Applications, SampTA 2017
A2 - Anbarjafari, Gholamreza
A2 - Kivinukk, Andi
A2 - Tamberg, Gert
PB - Institute of Electrical and Electronics Engineers Inc.
T2 - 12th International Conference on Sampling Theory and Applications, SampTA 2017
Y2 - 3 July 2017 through 7 July 2017
ER -