Rates of approximation and ergodic limits of regularized operator families
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 \[ \lim_{t\to\infty}{k(t) \over (k*a)(t)} = 0,\quad \sup_{t> 0} {k(t)(1*a)(t) \over (k*a)(t)} < \infty,\quad \sup_{t> 0} {(a*a*k)(t)\over (a*k)(t)} = \infty. \]
scientific article; zbMATH DE number 1924730