If ${f_k}_{k=1}^\infty$ is a sequence of Reimann Integral functions on [0,1] and $∫_0^1 |f_k(x) - f(x)|dx $ -> 0 as k -> $\infty$. Show $\hat{f}_k(n)$ -> $\hat{f}(n)$ converges uniformly in n as k -> $\infty$
So far a friend of mine told me that $L^1(T)$ convergence of $f_k$ implies uniform convergence of $\hat{f}_k$. But I don't see how that happens.
Any help would be appreciated.
Since the $f_k$ are continuous a.e.
$$|\hat{f}(n)-\hat{f}_k(n)|=\left | \int_0^1 (f(x)-f_k(x))e^{-inx}\right|\le \int_0^1|f(x)-f_k(x)|dx$$