Let $g(x)$ be a continuous real valued function defined on $[-a,1+a]$ where $a>0.$
Let $f(x)$ be a continuous real valued function, $f\left( x\right) \geq 0$ for $x\in \lbrack 0,1],$ equal to zero otherwise.
Let $g_{n}(x)$ be a continuous real valued function with domain $% [-a,1+a].$
Let $g_{n}\left( x\right) \rightarrow g\left( x\right) $ for any $x\in \lbrack -a,1+a],$ so that $g_{n}$ converges to $g$ pointwise.
For $\left\vert \Delta \right\vert define
$B_{n}(\Delta )=\int_{0}^{1}g_{n}(x+\Delta )f(x)dx$
$B(\Delta )=\int_{0}^{1}g(x+\Delta )f(x)dx$
Since $g_{n}(x+\Delta )$ converges pointwise to $g(x+\Delta ),$ $% B_{n}(\Delta )\rightarrow B(\Delta )$.
Question: under what conditions is the convergence uniform in $\Delta ?$
That is, can we find conditions such that for any $\varepsilon >0,\exists N$ such that $n>N\Rightarrow \left\vert B_{n}(\Delta )-B(\Delta )\right\vert <\varepsilon \forall \Delta \in [-a,a]$ ?