Fundamental theorems on regular almost-periodic functions

Abstract

The article covers important mathematical properties of uniformly almost-periodic functions and provides
proofs of the corresponding theorems. It is shown that a uniformly almost-periodic function is bounded
on the entire real axis and, as a consequence, the square of such a function also possesses the uniformly
almost-periodic property. It is also proved that these functions are uniformly continuous on the entire
real axis. Furthermore, theorems are established demonstrating that the sum and product of uniformly
almost-periodic functions retain the same property. The limit of a uniformly convergent sequence of such
functions is also shown to be a uniformly almost-periodic function. If the derivative of an almost-periodic
the integral itself is a uniformly almost-periodic function.
function is uniformly continuous on the entire real axis, then its derivative is also uniformly almost-periodic.
Finally, it is proved that if an indefinite integral of a uniformly almost-periodic function is bounded, then