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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 [itex]f: [-\pi,\pi)\rightarrow \mathbb{R}[/itex] a function (n-1) times continuously differentiable such that [itex]f^{(n-1)}(x)[/itex] is differentiable and continuous except maybe at a finite number of points. If [itex]|f^{(n)}(x)|\leq M[/itex] 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]<br /> <br /> Edit: [itex]|f^{(n-1)}(x)|\leq M[/itex] --> [itex]|f^{(n)}(x)|\leq M[/itex][/tex]
"Consider [itex]f: [-\pi,\pi)\rightarrow \mathbb{R}[/itex] a function (n-1) times continuously differentiable such that [itex]f^{(n-1)}(x)[/itex] is differentiable and continuous except maybe at a finite number of points. If [itex]|f^{(n)}(x)|\leq M[/itex] 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]<br /> <br /> Edit: [itex]|f^{(n-1)}(x)|\leq M[/itex] --> [itex]|f^{(n)}(x)|\leq M[/itex][/tex]
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