Fourier's Theorem/Lemma 2

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Lemma for Fourier's Theorem

Let $\psi$ be a real function defined on a half-open interval $\left({0 \,.\,.\, a}\right]$.

Let $\psi$ and its derivative $\psi'$ be piecewise continuous with one-sided limits on $\left({0 \,.\,.\, a}\right]$.

Let $\psi \left({u}\right)$ have a right-hand derivative at $u = 0$.


Then:

$\displaystyle \lim_{N \mathop \to \infty} \int_0^a \psi \left({u}\right) \frac {\sin N u} u \rd u = \frac \pi 2 \psi \left({0^+}\right)$

where $\psi \left({0^+}\right)$ denotes the limit of $\psi$ at $0$ from the right.


Proof

We have:

$\psi \left({u}\right) = \psi \left({0^+}\right) + \left({\psi \left({u}\right) - \psi \left({0^+}\right)}\right)$

from which:

$(1): \quad \displaystyle \int_0^a \psi \left({u}\right) \frac {\sin N u} u \rd u = \psi \left({0^+}\right) \int_0^a \frac {\sin N u} u \rd u + \int_0^a \phi \left({u}\right) \sin N u \rd u$

where:

$\phi \left({u}\right) = \dfrac {\psi \left({u}\right) - \psi \left({0^+}\right)} u$


Let $\xi = N u$.

Then:

\(\displaystyle \int_0^a \frac {\sin N u} u \rd u\) \(=\) \(\displaystyle \int_0^{N a} \frac {\sin \xi} \xi \rd \xi\) $\quad$ $\quad$
\(\displaystyle \) \(\to\) \(\displaystyle \int_0^\infty \frac {\sin \xi} \xi \rd \xi\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \frac \pi 2\) $\quad$ Integral to Infinity of $\dfrac {\sin \xi} \xi$ $\quad$


We have that $\psi \left({u}\right)$ is piecewise continuous with one-sided limits on $\left({0 \,.\,.\, a}\right]$.

Hence it follows that $\phi \left({u}\right) = \dfrac {\psi \left({u}\right) - \psi \left({0^+}\right)} u$ is also piecewise continuous with one-sided limits on $\left({0 \,.\,.\, a}\right]$.

We also have that $\psi \left({u}\right)$ has a right-hand derivative at $u = 0$.

It follows that $\phi \left({u}\right)$ is piecewise continuous with one-sided limits on $\left[{0 \,.\,.\, a}\right]$.


Thus from Lemma 1 for Fourier's Theorem:

$\displaystyle \lim_{N \mathop \to \infty} \int_0^a \phi \left({u}\right) \sin N u \rd u = 0$

and letting $N \to \infty$ in $(1)$ above:

$\displaystyle \lim_{N \mathop \to \infty} \int_0^a \psi \left({u}\right) \frac {\sin N u} u \rd u = \frac \pi 2 \psi \left({0^+}\right)$

$\blacksquare$


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