Toronto Math Forum
APM3462012 => APM346 Math => Final Exam => Topic started by: Djirar on December 20, 2012, 01:30:26 PM

Consider a $2\pi$periodic function $f$ with full Fourier series
$$
\sum_{n \in \mathbb{Z}} c_n e^{i n x}.
$$
Suppose that the Fourier coefficients decay fast enough to satisfy
$$
\sum_{n \in \mathbb{Z}} n \cdot c_n < 17.
$$
Prove that $f'$ is bounded.

I thought we were supposed to wait until Prof. Ivrii posted the problems...?
In any case, my solution to Problem 2 is attached.

My answer to question 2

I will leave this problem to grade to Prof. Colliander. None of the posted solutions satisfies me.
Proof. Consider
\begin{equation}
g(x)=\sum_{n=\infty}^\infty nc_n e^{inx}.
\label{eq1}
\end{equation}
Since $\sum_{n=\infty}^\infty nc_n e^{inx}=\sum_{n=\infty}^\infty n\cdot c_n\le M$, series (\ref{eq1}) converges uniformly and therefore one can integrate it termwise:
\begin{equation}
\int_0^x g(x)\,dx=\sum_{n=\infty}^\infty \int_0^x nc_n e^{inx}\,dx =\sum_{n=\infty}^\infty \int_0^x c_n \bigl(e^{inx}1)=f(x)f(0)
\label{eq2}
\end{equation}
Therefore $f(x)$ is differentiable and $f'(x)=g(x)$.
PS. You can differentiate series termwise if you get uniformly converging series. Nobody mentioned this.