The authors consider the classes and , where is either continuous with as , or weakly Eberlein almost periodic , with bounded continuous functions, or , Banach space; denotes any translation invariant closed linear subspace of containing all constants and with also , , , and such that is isometric; denotes a space containing only recurrent functions; . These classes subsume various generalized almost periodic and almost automorphic functions, e.g. almost periodic functions or uniformly continuous bounded Levitan - a.p. functions. It is shown that the in is direct, and are closed subspaces of and closed with respect to convolution with . If and is ergodic, then . When the Beurling spectrum of a uniformly continuous bounded has positive distance from 0, then if [resp. all differences , then [resp. ].
As an application, if is a bounded solution on of with for all real , then ; for systems where the matrix has no purely imaginary eigenvalues and , are suitably asymptotic , the existence of a unique solution for sufficiently small is shown. Similarly, a bounded solution of the quasilinear heat equation with initial data is again . (In theorem 4.6, p. 1146, instead of “(4.3)” it should read “(4.4) with and instead of ”).