A function $f:\mathbb{R}\to \mathbb{C}$ is said to be a trigonometric polynomial if it has the form $$f(x)=\sum_{k=-N}^Na_ke^{ib_kx},$$ where $a_k\in \mathbb{C}$ and $b_k\in \mathbb{R}$.
Can we find a function $g:\mathbb{R}\to \mathbb{C}$ such that $$\int_0^{+\infty}|g(x)|dx<+\infty,$$ and $g$ is the limit of a sequence of trigonomteric polynmials $f_n$ with respect to the norm $$||g||=\sup_{t\in\mathbb{R}}\int_t^{t+1}|g(x)|dx.$$ Note: I tried to look at that function $g$ defined to be zero everywhere except for triangular peaks of height $1$ and base $\dfrac{1}{n^2}$ centered on positive integers $k$, so the surface of each triangle is $\dfrac{1}{2n^2}$. It is a classic example of a function which is integrable because $\int_0^{+\infty}|g(x)|dx=\sum_n\frac{1}{n^2}<+\infty$, but does not have a limit in $+\infty$. I choose this function because I think it does have some sort of oscillations so there may be a hope to approximate it with trigonometric polynomials using the norm I mentioned. But I don't know if it is possible to find an approximating sequence.
Such sequence does not exist, if we assume the following Diophantine approximation result:
Lemma 1: If a trigonometric polynomial $f(x)=\sum_{k=-N}^{N}a_k e^{ib_k x}$ satisfies $|f(x_0)|=2c>0$ for some $x_0\in\mathbb{R}$, there exists an infinite number of disjoint intervals $I\subset\mathbb{R}^+$ with bounded width such that $\int_I|f(x)|\,dx\geq\eta c$, where $\eta$ is an absolute positive constant.
Corollary: No non-zero trigonometric polynomial belongs to $L^1(\mathbb{R}^+)$.
Let $d=\int_{0}^{1}|g(x)|\,dx$. Since $f_n\to g$ with respect to our "almost uniform" norm, for any $n$ big enough
$$\int_{0}^{1}|f_n(x)|\,dx \geq \frac{d}{2}$$ holds, hence we can take $c=\frac{d}{4}$ for any $f_n$. Assuming that $g\in L^1(\mathbb{R}^+)$, then: $$\lim_{t\to +\infty}\int_{t}^{t+1}|g(t)|\,dt=0,$$ so $$\lim_{n\to +\infty}\lim_{t\to +\infty}\int_{t}^{t+1}|g(x)-f_n(x)|\,dx = 0,$$ but the innermost limit cannot be zero due to Lemma 1.