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How to show that if $f$ is bounded and monotonic on $[a,b]$ then $|\hat{f}(n)| \leq \frac{c}{n}$, i.e the Fourier coefficients are bounded?

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You claim much more than just that the Fourier coefficients are bounded, you want to prove that they decrease in a controlled way as $n \rightarrow \infty$.

The best way to answer this question depends strongly on your knowledge about integration. If you are familiar with the Riemann-Stieltjes integral the claim follows from the following formula: if $g$ is continuously differentiable and $f$ monotonic on $[a,b]$, then $$ \int_a^b f(x) g'(x) dx = g(b)f(b) - g(a)f(a) - \int_a^b g(y)df(y) $$ If you apply this formula to $g(y) = -\frac{1}{in} e^{-iny}$ (out of laziness I assume here $[a,b] = [-\pi,\pi]$) then the left hand side is just a constant times $\hat{f}(n)$ and the rhs is bounded by some constant times $\frac{1}{n}$, and you are done.

If you are not familiar with the Riemann-Stieltjes integral (or don't know that integration by parts formula) then I guess it is a bit more work.

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You might want to use the Second Mean Value Theorem for integral. It is quite straight forward. No need to appeal to Riemann-Stieltjes. – user231543 Apr 23 '15 at 15:34

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