# Besicovitch almost periodic functions with seminorm zero

Consider the set of all trigonometrical polynomials of the form $$P(x)=\sum_{j=1}^n a_n e^{ix\cdot\xi_n}$$, where $$\xi_n\in\mathbb{R}^d$$. A function is said to be almost periodic in the sense of Bohr if it is a uniform limit of trigonometrical polynomials. Define mean value of a function $$f\in L^1_{loc}(\mathbb{R}^d)$$ to be the number $$\mathcal{M}(f)=\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^T f(x)\,dx.$$ Then, a function $$u$$ is said to be almost periodic in the sense of Besicovitch if there is a sequence of trigonometrical polynomials $$P_n$$ such that $$\mathcal{M}(|u-P_n|^2)\to 0$$ as $$n\to\infty$$. In most texts, it is mentioned that the quantity $$(\mathcal{M}(|u|^2))^{1/2}$$ is a semi-norm on the set of all Besicovitch almost periodic functions, however I have not found any text which would provide an example of a non-zero Besicovitch almost periodic function $$u$$ with $$\mathcal{M}(|u|^2)=0$$.

Is anyone aware of such an example? I am not able to construct such an example and it is certainly not true for trigonometrical polynomials. Of course, I mean examples which are not zero on a set of positive measure.

• $$f(x)=\left\{ \begin{array}{lc} 1 \mbox{ if } 2^n < x <2^{n}+1 \\ 0 &\mbox{otherwise} \end{array}\right.$$ and $P_n=0$. Jun 13, 2020 at 18:55

Take any compactly suported $$L^2$$ function $$f$$ and set $$P_n=0$$, then $$\lim_{n\rightarrow \infty} \mathcal{M}(\vert f - P_n \vert^2) = \mathcal{M}(\vert f \vert^2) =0$$ thus, $$f$$ is Besicovitch almost periodic such that its seminorm vanishes.