Convergence at the origin of integrated semigroups

Vincent Cachia. "Convergence at the origin of integrated semigroups." Studia Mathematica 187.3 (2008): 199-218. <http://eudml.org/doc/284501>.

@article{VincentCachia2008,
abstract = {We study a classification of κ-times integrated semigroups (for κ > 0) by their (uniform) rate of convergence at the origin: $||S(t)|| = (t^\{α\})$ as t → 0 (0 ≤ α ≤ κ). By an improved generation theorem we characterize this behaviour by Hille-Yosida type estimates. Then we consider integrated semigroups with holomorphic extension and characterize their convergence at the origin, as well as the existence of boundary values, by estimates of the associated holomorphic semigroup. Various examples illustrate these results. The particular case α = κ, which corresponds to the notions of Riesz means or tempered integrated semigroups, is of special interest; as an application, it leads to an integrated version of Euler’s exponential formula.},
author = {Vincent Cachia},
journal = {Studia Mathematica},
keywords = {integrated semigroups; holomorphic semigroups; exponential formula},
language = {eng},
number = {3},
pages = {199-218},
title = {Convergence at the origin of integrated semigroups},
url = {http://eudml.org/doc/284501},
volume = {187},
year = {2008},
}

TY - JOUR
AU - Vincent Cachia
TI - Convergence at the origin of integrated semigroups
JO - Studia Mathematica
PY - 2008
VL - 187
IS - 3
SP - 199
EP - 218
AB - We study a classification of κ-times integrated semigroups (for κ > 0) by their (uniform) rate of convergence at the origin: $||S(t)|| = (t^{α})$ as t → 0 (0 ≤ α ≤ κ). By an improved generation theorem we characterize this behaviour by Hille-Yosida type estimates. Then we consider integrated semigroups with holomorphic extension and characterize their convergence at the origin, as well as the existence of boundary values, by estimates of the associated holomorphic semigroup. Various examples illustrate these results. The particular case α = κ, which corresponds to the notions of Riesz means or tempered integrated semigroups, is of special interest; as an application, it leads to an integrated version of Euler’s exponential formula.
LA - eng
KW - integrated semigroups; holomorphic semigroups; exponential formula
UR - http://eudml.org/doc/284501
ER -