Duoandikoetxea's Fourier Analysis, on page 59 (Corollary 3.7) says that:
\begin{equation} \lim_{R \rightarrow \infty}\big\|S_{R}\,f - f\big\|_{p} = 0 \end{equation}
for $1<p<\infty$, where $S_{R}\,f \:$ is such that $\widehat{S_{R}\,f}\left(y\right)$ = $\chi_{(-R,R)}\left(y\right) \,\hat{f}(y)$. He also proves that there is a constant $C_{p}$ such that $||\,S_{R}\,f\,||_{p} \le C_{p}\|\,f\|_{p}$, and that this $C_{p}$ doesn't depend on $R$.
After that he says that this is not true when $p=1$, but he doesn't give a counterexample. What function would contradict this when $p=1$?
Another thing, he obtains the limit above as a consequence of $\|\,S_{R}\,f\,\|_{p} \leq C_{p}\,\|f\|_{p}$. I can prove the limit above for all $f \in L^{p}$ if I know that it is valid for a dense subset, let's say, $S(\mathbb{R})$ (Schwartz class) or $C^{\infty}_{c}$ by approximating $f$ using such functions. How do I prove the limit above for $f$ in any of these spaces?
Suppose we have that $\|S_R f\|_{p}\le C_p\|f\|_{p}$ for some constant $C_p >0$, $p\in (0,\infty)$. Since Schwartz functions are dense in $L^p$, for every $\varepsilon >0$, there exists $g\in\mathcal{S}$ such that $\|f-g\|_{p}<\varepsilon$.
\begin{align} \|S_R f-f\|_{p} & \le\|S_R f-S_R g\|_{p}+\|S_R g-g\|_{p}+\|g-f\|_{p}\\ & \le (1+C_p)\|f-g\|_{p}+\|S_R g-g\|_{p} \end{align} Also, \begin{align} g(x) & =\lim_{R\rightarrow\infty}\int_{|\xi|<R} \hat g(\xi)e^{2\pi ix\xi}d\xi\\ & =\lim_{R\rightarrow\infty} S_R g \end{align}
So for $\varepsilon >0$ as above, there exists a $\tilde R>0$ such that $|S_R g-g|<\varepsilon$ for all $R>\tilde R$. Thus, for all $R>\tilde R$, $\|S_R f-f\|_{p}<C\varepsilon$. But $\varepsilon$ is arbitary.