Fourier's Theorem/Lemma 2

Lemma for Fourier's Theorem

Let $\psi$ be a real function defined on a half-open interval $\hointl 0 a$.

Let $\psi$ and its derivative $\psi'$ be piecewise continuous with one-sided limits on $\hointl 0 a$.

Let $\map \psi u$ have a right-hand derivative at $u = 0$.

Then:

$\ds \lim_{N \mathop \to \infty} \int_0^a \map \psi u \frac {\sin N u} u \rd u = \frac \pi 2 \map \psi {0^+}$

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

Proof

We have:

$\map \psi u = \map \psi {0^+} + \paren {\map \psi u - \map \psi {0^+} }$

from which:

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

where:

$\map \phi u = \dfrac {\map \psi u - \map \psi {0^+} } u$

Let $\xi = N u$.

Then:

\(\ds \int_0^a \frac {\sin N u} u \rd u\)

\(=\)

\(\ds \int_0^{N a} \frac {\sin \xi} \xi \rd \xi\)

\(\ds \)

\(\to\)

\(\ds \int_0^\infty \frac {\sin \xi} \xi \rd \xi\)

\(\ds \)

\(=\)

\(\ds \frac \pi 2\)

Integral to Infinity of $\dfrac {\sin \xi} \xi$

We have that $\map \psi u$ is piecewise continuous with one-sided limits on $\hointl 0 a$.

Hence it follows that $\map \phi u = \dfrac {\map \psi u - \map \psi {0^+} } u$ is also piecewise continuous with one-sided limits on $\hointl 0 a$.

We also have that $\map \psi u$ has a right-hand derivative at $u = 0$.

It follows that $\map \phi u$ is piecewise continuous with one-sided limits on $\hointl 0 a$.

Thus from Lemma 1 for Fourier's Theorem:

$\ds \lim_{N \mathop \to \infty} \int_0^a \map \phi u \sin N u \rd u = 0$

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

$\ds \lim_{N \mathop \to \infty} \int_0^a \map \psi u \frac {\sin N u} u \rd u = \frac \pi 2 \map \psi {0^+}$

$\blacksquare$

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Sources

• 1961: I.N. Sneddon: Fourier Series ... (previous) ... (next): Chapter Two: $\S 2$. Some Important Limits: Lemma $(2)$