Let $f,g:\mathbb{R}\rightarrow\mathbb{R}$ be two continuosly differentiable functions. For a work I need to prove that $$\sum_{-n\leq k\leq n}f(k)g(nx-k)$$ where $n\in\mathbb{N}^{+}$ and $x\in\left[-1,1\right]$, converges as $n\rightarrow+\infty$ uniformly with respect to $x$. For technical reason, I'm able to prove this fact if I consider $$\sum_{n(x-1)\leq k\leq n(x+1)}f(nx-k)g(k)$$ but I'm not able to estimate the error $$\left|\sum_{-n\leq k\leq n}f(k)g(nx-k)-\sum_{n(x-1)\leq k\leq n(x+1)}f(nx-k)g(k)\right|.\tag{1}$$ This operation recalls, in some sense, the following property of the discrete convolution $$\sum_{k\in\mathbb{Z}}f(k)g(x-k)=\sum_{k\in\mathbb{Z}}f(x-k)g(k)$$ but for a finite sum I don't know what happens.
Question: is it known an estimation for $(1)$?