Eberlein almost periodic functions that are not pseudo almost periodic

TL;DR

The paper constructs specific Eberlein almost periodic functions in Hilbert spaces that challenge existing notions by not being pseudo almost periodic, answering open questions in the field.

Contribution

It provides explicit examples of Eberlein almost periodic functions that are not pseudo almost periodic, clarifying the distinctions between these classes.

Findings

Constructed Eberlein almost periodic functions with non-ergodic norms

Demonstrated functions where the norm is Eberlein almost periodic but the functions are not pseudo almost periodic

Showed the failure of the Parseval equation for these functions

Abstract

We construct Eberlein almost periodic functions $ f_j : J \to H$ so that $||f_1(\cdot)||$ is not ergodic and thus not Eberlein almost periodic and $||f_2(.)||$ is Eberlein almost periodic, but $f_1$ and $f_2$ are not pseudo almost periodic, the Parseval equation for them fails, where $J=\r_+$ or $\r$ and $H$ is a Hilbert space. This answers several questions posed by Zhang and Liu [18].