I am trying to formulate a problem mathematically for which I have to mathematically formulate pseudo-periodic time series. According to this research paper, Almost-periodic and pseudo-periodic signals are similar.
I am not able to completely understand the below mathematical definitions of almost periodic functions. I tried to write them in simple terms below. However, I wanted your help to correct me if my interpretation is wrong.
Layperson's summary of almost-periodic functions in terms of time series
Take very small repeated segment's (referred as sequence in definition) of a modified time series (I think it's called a translation) with a approximate period denoted by epsilon where epsilon or period is greater than zero. If the similarity between big chunk of these consecutive sequences (referred as sub-sequences in definition) in terms of period and shape is less than epsilon than the whole time series is called pseudo-periodic time series.
Research paper mathematically defined almost-periodic signal by following the original definition proposed by H. Bohr (1947):
However, on wikipedia page the same equation is shown by Bohr (1925) and Bochner (1926):
