Unraveling the Mystery: Solving a Tricky Fourier Analysis Problem

[quasar987]

This seemingly not-so-harsh math problem has me stumped. I tried solving it every free minute I had this weekend but no trails or any combination of them led me anywhere happy. The little ba$tard goes as follow:

"Consider $f: [-\pi,\pi)\rightarrow \mathbb{R}$ a function (n-1) times continuously differentiable such that $f^{(n-1)}(x)$ is differentiable and continuous except maybe at a finite number of points. If $|f^{(n)}(x)|\leq M$ except maybe at the points of discontinuity, show that the coefficients of the development of f in a complex Fourier serie satisfy

[tex]|c_r|\leq M/r^n, \ \forall r \neq 0[/itex]

Edit: $|f^{(n-1)}(x)|\leq M$ --> $|f^{(n)}(x)|\leq M$

 

[CarlB]

Well, with that factor of $r^n$ it sure smells like an integration by parts is involved.

Carl

 

[quasar987]

Thanks for the reply CarlB, I didn't see it that way. I'll try to see what I can do with integration by parts...

 

[quasar987]

Integration by parts it was!

Whenever you need a hug CarlB, I,m here for you.