5.1.B. Discussion: pointwise convergence of Fourier integrals and series

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5.1.B. Discussion: pointwise convergence of Fourier integrals and series

Recall Theorem 4.4.2: Let $f$ be a piecewise continuously differentiable function. Then the Fourier series $$\frac{a_0}{2}+\sum_{n=1}^\infty \Bigl(a_n\cos \bigl(\frac{\pi n x}{l}\bigr) + a_n\cos \bigl(\frac{\pi n x}{l}\bigr)\Bigr) \label{eq-5.1B.1}$$ converges to

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

Exactly the same statement holds for Fourier Integral in the real form $$\int_0^\infty \Bigl(A( k ) \cos ( k x) + B( k )\sin ( k x)\Bigr)\,d k \label{eq-5.1B.2}$$ where $A( k )$ and $B( k )$ 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-5.1B.3}\\ \int_{-\infty}^\infty C( k )e^{i k x}\,d k . \label{eq-5.1B.4} \end{gather}

Why and what remedy do we have? If we consider definition of the partial sum of (\ref{eq-5.1B.1}) and then rewrite in the complex form and similar deal with (\ref{eq-5.1B.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-5.1B.5}\\ \lim_{N\to \infty} \int_{-N}^N C( k )e^{i k x}\,d k \label{eq-5.1B.6}. \end{gather} Meanwhile convergence in (\ref{eq-5.1B.3}) and (\ref{eq-5.1B.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-5.1B.7}\\ \lim_{M,N\to \infty} \int_{-M}^N C( k )e^{i k x}\,d k \label{eq-5.1B.8} \end{gather} where $M,N$ tend to $\infty$ independently. So the remedy is simple: understand convergence as in (\ref{eq-5.1B.5}), (\ref{eq-5.1B.6}) rather than as in (\ref{eq-5.1B.7}), (\ref{eq-5.1B.8}).

For integrals such limit is called principal value of integral and is denoted by \begin{equation*} \operatorname{pv}\int_{-\infty}^\infty G( k )\,d k . \end{equation*} Also similarly is defined the principal value of the integral, divergent in the finite point \begin{equation*} \operatorname{pv}\int_{a}^b G( k )\,d k := \lim_{\varepsilon\to +0} \Bigl(\int_a^{c-\varepsilon}G( k )\,d k + \int_{c+\varepsilon}^bG( k )\,d k \Bigr) \end{equation*} if there is a singularity at $c\in (a,b)$. Often instead of pv is used original (due to Cauchy) vp (valeur principale) and some other notations.

This is more general than the improper integrals studied in the end of Calculus I (which in turn generalize Riemann integrals). Those who took Complex Variables encountered such notion.