Author Topic: Problem 4  (Read 10479 times)

Victor Ivrii

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Re: Problem 4
« Reply #15 on: November 08, 2012, 04:32:56 PM »
Fanxun, I decided that you deserve a karma, for raising a good question. In Lecture 15 we proved Theorem 2:
 
Theorem 2. Let $f$ be a piecewise continuously differentiable function. Then the Fourier series
\begin{equation}
\frac{a_0}{2}+\sum_{n=1}^\infty \Bigl(a_n\cos \bigl(\frac{\pi n x}{l}\bigr) + a_n\cos \bigl(\frac{\pi n x}{l}\bigr)\Bigr)
\label{eq-1}
\end{equation}
converges to

(b) $\frac{1}{2}\bigl(f(x+0)+f(x-0)\bigr)$  if $x$ is internal point and $f$ is discontinuous at $x$.

Exactly the same statement holds for Integral Fourier
\begin{equation}
\int_0^\infty \Bigl(A(\omega) \cos (\omega x) + B(\omega)\sin (\omega x)\Bigr)\,d\omega
\label{eq-2}
\end{equation}
where $A(\omega)$ and $B(\omega)$ are $\cos$- and $\sin$-Fourier transforms.

None of them however holds for Fourier series or Fourier Integral in the complex form:
\begin{gather}
\sum_{n=-\infty}^\infty c\_n e^{i\frac{\pi n x}{l}},\label{eq-3}\\
\int_{-\infty}^\infty C(\omega)e^{i\omega x}\,d\omega.\label{eq-4}
\end{gather}

Why and what remedy do we have? If we consider definition of the partial sum of (\ref{eq-1}) and then rewrite in the complex form  and similar deal with (\ref{eq-4}) we see that in fact we should look at
\begin{gather}
\lim_{N\to \infty} \sum_{n=-N}^N c_n e^{i\frac{\pi n x}{l}},\label{eq-5}\\
\lim_{N\to \infty} \int_{-N}^N C(\omega)e^{i\omega x}\,d\omega\label{eq-6}.
\end{gather}

Meanwhile convergence in (\ref{eq-3}) and (\ref{eq-4}) means more than this:
\begin{gather}
\lim_{M,N\to \infty} \sum_{n=-M}^N c_n e^{i\frac{\pi n x}{l}},\label{eq-7}\\
\lim_{M,N\to \infty} \int_{-M}^N C(\omega)e^{i\omega x}\,d\omega\label{eq-8}
\end{gather}
where $M,N$ tend to $\infty$ independently. So the remedy is simple: understand convergence as in (\ref{eq-5}), (\ref{eq-6}) rather than as in (\ref{eq-7}), (\ref{eq-8}) . For integrals such limit is called essential value of integral and is denoted by
\begin{equation*}
\operatorname{ess}-\int_{-\infty}^\infty G(\omega)\,d\omega
\end{equation*}
or
\begin{equation*}
\operatorname{vrai}-\int_{-\infty}^\infty G(\omega)\,d\omega
\end{equation*}

BTW similarly is defined \begin{equation*}
\operatorname{ess}-\int_{a}^b G(\omega)\,d\omega:= \lim_{\epsilon\to +0} \Bigl(\int_a^{c-\epsilon}G(\omega)\,d\omega+
\int_{c+\epsilon}^bG(\omega)\,d\omega\Bigr)
\end{equation*}
if there is a singularity at $c\in (a,b)$.

This is more general than improper integrals studied in the end of Calculus I. Those who took Complex Variables encountered such notion.
« Last Edit: November 08, 2012, 04:36:11 PM by Victor Ivrii »