Toronto Math Forum

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

Title: problem 2
Post 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.

Title: Re: problem 2
Post by: Chen Ge Qu on December 20, 2012, 01:35:34 PM

I thought we were supposed to wait until Prof. Ivrii posted the problems...?

In any case, my solution to Problem 2 is attached.

Title: Re: problem 2
Post by: Pei Zhou on December 20, 2012, 04:49:39 PM

My answer to question 2

Title: Re: problem 2
Post by: Victor Ivrii on December 22, 2012, 01:02:24 PM

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{eq-1}
\end{equation}
Since $\sum_{n=-\infty}^\infty |nc_n e^{inx}|=\sum_{n=-\infty}^\infty |n|\cdot |c_n|\le M$, series (\ref{eq-1}) 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{eq-2}
\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.