Besicovitch almost periodic function

Remarks

The quantity $\norm{\cdot}_{B^p}$ is a seminorm, not a norm. It vanishes on every function in the Lebesgue space $L^p(\R)$, and the space of $B^p$ classes is obtained by identifying functions $f, g$ with $\norm{f - g}_{B^p} = 0$. Every Bohr almost periodic function is $B^p$ almost periodic for every $p$, since $\norm{g}_{B^p} \le \sup \abs{g}$. For $p = 2$, each $B^2$ almost periodic $f$ has a mean value of $\abs{f}^2$ and Fourier coefficients $a(\lambda)$, nonzero for at most countably many $\lambda$, with $\lim_{T \to \infty} \frac{1}{2T} \int_{-T}^{T} \abs{f}^2 = \sum_\lambda \abs{a(\lambda)}^2$.

Example: $f = 1_{[0,1]}$ is $B^p$ almost periodic for every $p$. Indeed $\frac{1}{2T} \int_{-T}^{T} \abs{f}^p = \frac{1}{2T}$ for $T \ge 1$, so $\norm{f - 0}_{B^p} = 0$ and the polynomial $P = 0$ works. This $f$ is not continuous, so it is not Bohr almost periodic.

Non-example: $f(x) = x$ is not $B^p$ almost periodic. Every generalized trigonometric polynomial $P$ is bounded, say by $C$, so $\abs{f - P} \ge \abs{x} - C$ and $\norm{f - P}_{B^p} = \infty$.