Inequality on $L_1$ norms of tirgonometric polynomials generated with a smooth function

Question

Let $\varphi\in C_0^\infty(\mathbb R)$ and for $n\ge1$ $$ f_n(x)=\sum_{k=-\infty}^\infty \varphi(k/n)e^{i k x}. $$ I seem to remember that there is an inequality $\|f_n\|_{L_1(\mathbb T)}\le C$, where $C$ does not depend on $n$. Some lax reasonings give $\|f_n\|_{L_1(\mathbb T)}\le C\|\tilde\varphi\|_{L_1(\mathbb R)}$. Is there a reference for this inequality or similar things?