Find the limit of Fourier coefficients

Question

Suppose that $f(x)$ is on continuously differentiable function, with bounded derivative, defined on $[0,1]$ such that $$ \lim\limits_{x\to1^-}f(x)-\lim\limits_{x\to0^+}f(x)=1 $$ The Fourier coefficients of f are defined by $$ \hat{f}(n)=\int_0^1f(x)e^{-2\pi inx}dx. $$ Prove that, as $n \to ±\infty$, we have: $$ \hat{f}(n)=\frac{i}{2\pi n}+o\left(\frac{1}{n}\right) $$

Answer 1

Hint: An integration by parts shows that $$\hat f(n)=\frac1{-2\pi i n} +\frac1{2\pi in}\widehat{f'}(n).$$