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Zbl 1032.47024
Rates of approximation and ergodic limits of regularized operator families.
(English)
[J] J. Approximation Theory 122, No.1, 42-61 (2003). ISSN 0021-9045

If $a$ and $k$ are Laplace transformable functions on $[0,\infty)$ and $A$ is a closed linear operator, the $(a,k)$ regularized family generated by $A$ is the strongly continuous function $R$ satisfying $$\widehat k(\lambda)(I-\widehat a(\lambda)A)^{-1} = \int_0^\infty e^{-\lambda s}R(s) ds.$$ The authors study the behavior as $t\to \infty$ of the family of operators $$A_tx = {{1}\over{(k*a)(t)}} \int_0^t a(t-s)R(s)x ds.$$ By choosing $a$ and $k$ appropriately, the family $\{R(t)\}_{t\geq 0}$ corresponds to an $n$-times integrated semigroup, resolvent family, or cosine family, etc. The basic assumptions on $a$ and $k$ that make it possible to draw conclusions on the asymptotic behavior of $A_t$ is that $a(t)$ is positive, $k(t)$ is positive and decreasing; the most important additional hypotheses employed are $$\lim_{t\to\infty}{k(t) \over (k*a)(t)} = 0,\quad \sup_{t> 0} {k(t)(1*a)(t) \over (k*a)(t)} < \infty,\quad \sup_{t> 0} {(a*a*k)(t)\over (a*k)(t)} = \infty.$$
[Gustaf Gripenberg (Hut)]
MSC 2000:
*47D06 One-parameter semigroups and linear evolution equations
45J05 Integro-ordinary differential equations
47D62 Integrated semigroups

Keywords: strong ergodicity; uniform ergodicity; convoluted semigroup; integrated semigroup; resolvent

Cited in: Zbl 1156.47045

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