Decay rate of fourier coefficients of function which has a jump discontinuity

Question

I have read this statement in a lecturer notes of Fourier analysis: "If a function has a jump discontinuity, then its Fourier coefficients decay like $ \frac{1}{n} $." But, I have not been able to prove it. Could someone provide an explanation to verify this statement. Thank you.

Answer 1

If $f$ is a $1$-periodic piecewise $C^1$ function with finitely many jump discontinuites at $a_l$ of height $b_l$ then integrating by parts $$c_n(f)=\int_0^1 f(x)e^{-2i\pi nx}dx$$ gives $$c_n(f) = \frac{c_n(f')}{2i\pi n}+\frac{\sum_{l=1}^L b_l e^{2i\pi a_l n}}{2i\pi n}$$ where $f'$ is the piecewise continuous $1$-periodic function obtained by differentiating $f$ away from the discontinuities, since $f'\in L^2$ then $\sum_n |c_n(f')|^2=\|f'\|_{L^2[0,1]}^2$ implies that $c_n(f')\to 0$.