THE LOCAL ERGODIC THEOREM FOR GROUPS OF UNITARY OPERATORS AND SECOND ORDER STATIONARY PROCESSES

Abstract

Let , be a strongly continuous unitary group in , where is a -finite measure. The local ergodic theorem is the relation for . It is shown that this relation is not satisfied for all and . Necessary and sufficient conditions are obtained for the local ergodic theorem in terms of properties of the spectral measure , where is the resolution of the identity corresponding to the group . In particular, (1) is satisfied if the integral converges. Generalizations to multiparameter groups and homogeneous random fields are given. Bibliography: 10 titles.