Non-Optimal Rates of Ergodic Limits and Approximate Solutions

This paper is concerned with non-optimal rates of convergence for two processes {A?} and {B?}, which satisfy ?A??=O(1),B?A?AB?=I?A?, ?AA??=O(e(?)), whereAis a closed operator ande(?)?0. Under suitable conditions, we describe, in terms ofK-functionals, thosex(resp.y) for which the non-optimal convergence rate of {A?x} (resp. {B?y}) is of the orderO(f(?)), wherefis a function satisfyinge(?)?f(?)?0. In case thatf(?)/e(?)?∞, the sharpness of the non-optimal rate of {A?x} is equivalent to thatAhas non-closed range. The result provides a unified approach to dealing with non-optimal rates for many particular mean ergodic theorems and for various methods of solving the equationAx=y. We discuss in particular applications to?-times integrated semigroups,n-times integrated cosine functions, and tensor product semigroups.

[1]  Sen-Yen Shaw.  Convergence rates of ergodic limits and approximate solutions , 1993 .

[2]  Sen-Yen Shaw,et al.  Mean ergodic theorems and linear functional equations , 1989 .

[3]  H. Fattorini,et al.  Ordinary differential equations in linear topological space. II. , 1969 .

[4]  H. Berens.  Interpolationsmethoden zur Behandlung von Approximationsprozessen auf Banachräumen , 1968 .

[5]  S. Shaw,et al.  On the equations Ax = q and SX − XT = Q , 1988 .

[6]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[7]  P. Butzer,et al.  The mean ergodic theorem for cosine operator functions with optimal and non-optimal rates , 1995 .

[8]  Matthias Hieber.  Laplace Transforms and α-Times Integrated Semigroups , 1991 .

[9]  R. Showalter,et al.  Convergence rates of ergodic limits for semigroups and cosine functions , 1978 .

[10]  Jung-Chan Chang,et al.  Rates of approximation and ergodic limits of resolvent families , 1996 .

[11]  Sen-Yen Shaw,et al.  Uniform convergence of ergodic limits and approximate solutions , 1992 .

[12]  S. Shaw.  Solvability of linear functional equations in Lebesgue spaces , 1990 .

[13]  W. Arendt.  Vector-valued laplace transforms and cauchy problems , 2002 .

[14]  Resonance principles with applications to mean ergodic theorems and projection operators , 1995 .

[15]  Y. Chen.  [The change of serum alpha 1-antitrypsin level in patients with spontaneous pneumothorax]. , 1995, Zhonghua jie he he hu xi za zhi = Zhonghua jiehe he huxi zazhi = Chinese journal of tuberculosis and respiratory diseases.

[16]  Abstract ergodic theorems , 1963 .

[17]  O. Davydov.  Some condensation theorems for semigroup operators , 1993 .

[18]  M. Sova,et al.  Cosine operator functions , 1966 .