Integral of an alpha unpredictable function

Abstract

The present paper is devoted to the study of integral properties of alpha unpredictable functions. The problem of invariance of recurrence under integration is one of the most challenging and interesting in the theory of functions. Let us start with periodicity. Next, the problem was solved for quasiperiodic, almost periodic, and Poisson stable functions. All of the problems were subjected to the condition of bounded integrals. The alpha-unpredictable functions are the endpoint in the row of recurrent functions. The class is the cross-border point from regularity to chaos in the dynamics presented through the elements of the row. In the present research, we finalized the proof that any theoretical functional recurrence is invariant with respect to integration, provided the result is bounded.