TY - JOUR
T1 - Divergence Behavior of Sequences of Linear Operators with Applications
AU - Boche, Holger
AU - Mönich, Ullrich J.
N1 - Publisher Copyright:
© 2018, Springer Science+Business Media, LLC, part of Springer Nature.
PY - 2019/4/15
Y1 - 2019/4/15
N2 - In this paper we study the spaceability of divergence sets of sequences of bounded linear operators on Banach spaces. For Banach spaces with the s-property, we can give a sufficient condition that guarantees the unbounded divergence on a set that contains an infinite dimensional closed subspace after the zero element has been added. This generalizes the classical Banach–Steinhaus theorem which implies that the divergence set is a residual set. We further prove that many important spaces, e.g., ℓ p , 1 ≤ p< ∞, C[0, 1], L p , 1 < p< ∞, as well as Paley–Wiener and Bernstein spaces, have the s-property. Finally, consequences for the convergence behavior of sampling series and system approximation processes are shown.
AB - In this paper we study the spaceability of divergence sets of sequences of bounded linear operators on Banach spaces. For Banach spaces with the s-property, we can give a sufficient condition that guarantees the unbounded divergence on a set that contains an infinite dimensional closed subspace after the zero element has been added. This generalizes the classical Banach–Steinhaus theorem which implies that the divergence set is a residual set. We further prove that many important spaces, e.g., ℓ p , 1 ≤ p< ∞, C[0, 1], L p , 1 < p< ∞, as well as Paley–Wiener and Bernstein spaces, have the s-property. Finally, consequences for the convergence behavior of sampling series and system approximation processes are shown.
KW - Banach–Steinhaus theorem
KW - Paley–Wiener spaces
KW - Sampling series
KW - Spaceability
KW - System and signal approximation
UR - http://www.scopus.com/inward/record.url?scp=85040691678&partnerID=8YFLogxK
U2 - 10.1007/s00041-018-9594-6
DO - 10.1007/s00041-018-9594-6
M3 - Article
AN - SCOPUS:85040691678
SN - 1069-5869
VL - 25
SP - 427
EP - 459
JO - Journal of Fourier Analysis and Applications
JF - Journal of Fourier Analysis and Applications
IS - 2
ER -