Ergodic theorems and approximation theorems with rates
The paper devoted to the study of some strong ergodic theorems, uniform ergodic theorems and ergodic theorems with rates. It is diveded into five sections. The first section is an introduction. In the second section it is given a review of some results in connections with \(A\)-ergodic nets, which were firstly introduced in [\textit{S.-Y. Shaw}, J. Funct. Anal. 87, No. 2, 428-441 (1989; Zbl 0704.47006)]. In this section for \(A\)-ergodic nets strong ergodic theorems, uniform ergodic theorems and ergodic theorems with rates are formulated. In the third section it is defined an approximation process and \(A\)-regularized approximation process on a Banach space \(X\). For these processes optimal convergence, non-optimal convergence and sharpness theorems are proved. In the forth section it is introduced, more exectly generalized \(r\)-times integrated solution family for the Volterra equation: \[ u(t)=f(t)+\int^{t}_{0}a(t-s)Au(s) ds, \quad t\geq 0, \] for \(f\in C([0;\infty);X)\), where \(A\) is a closed linear operator on \(X\) and \(a\in L^{1}_{\text{loc}}(\mathbb{R}^+)\) is a positive kernel. The notion of an \(r\)-times integrated solution family is an extention of the concept of \(n\)-times integrated semigroups and \(n\)-times integrated cosine functions. As applications of the general results to be formulated in sections 2 and 3, for \(r\)-times integrated solution family it is proved uniform ergodic theorem, strong ergodic theorem with rates. In section 5 for \(r\)-times integrated solution family local ergodic theorems with rates are proved. There are also established approximationsproperties of theorems given in section 4.
scientific article; zbMATH DE number 1547610