Fourier's Theorem

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Theorem

Let $\alpha \in \R$ be a real number.

Let $f \left({x}\right)$ be a real function which is defined and bounded on the interval $\left[{\alpha \,.\,.\, \alpha + 2 \pi}\right]$.

Let $f$ satisfy the Dirichlet conditions on $\left[{\alpha \,.\,.\, \alpha + 2 \pi}\right]$:

$(\mathrm D 1): \quad f$ is absolutely integrable.
$(\mathrm D 2): \quad f$ has a finite number of local maxima and local minima.
$(\mathrm D 3): \quad f$ has a finite number of discontinuities, all of them finite.


Outside the interval $\left[{\alpha \,.\,.\, \alpha + 2 \pi}\right]$, let $f$ be periodic and defined such that:

$f \left({x}\right) = f \left({x + 2 \pi}\right)$


Let $f$ be defined by the Fourier series:

$(1): \quad \displaystyle \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty \left({a_n \cos n x + b_n \sin n x}\right)$

such that:

$\displaystyle a_n = \dfrac 1 \pi \int_\alpha^{\alpha + 2 \pi} f \left({x}\right) \cos n x \rd x$
$\displaystyle b_n = \dfrac 1 \pi \int_\alpha^{\alpha + 2 \pi} f \left({x}\right) \sin n x \rd x$


Then for all $a \in \R$, $(1)$ converges to the sum:

$\displaystyle \frac 1 2 \left({\lim_{x \mathop \to a^+} f \left({x}\right) + \lim_{x \mathop \to a^-} f \left({x}\right)}\right)$

where the $\lim$ symbols denote the limit from the right and the limit from the left.


Proof

Lemma 1

Let $\psi$ be a real function defined on a closed interval $\left[{a \,.\,.\, b}\right]$.

Let $\psi$ be piecewise continuous with one-sided limits on $\left[{a \,.\,.\, b}\right]$.

Then:

$\displaystyle \lim_{N \mathop \to \infty} \int_a^b \psi \left({u}\right) \sin N u \rd u = 0$


Lemma 2

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.


Lemma 3

Let $\psi$ be a real function defined on an open interval $\left({a \,.\,.\, b}\right)$.

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

Let $\psi \left({u}\right)$ have both right-hand derivative and left-hand derivative at a point $u = x$ where $x \in \left({a \,.\,.\, b}\right)$.


Then:

$\displaystyle \lim_{N \mathop \to \infty} \int_a^b \psi \left({u}\right) \frac {\sin N \left({u - x}\right)} {u - x} \rd u = \frac \pi 2 \left({\psi \left({x^+}\right) + \psi \left({x^-}\right)}\right)$

where:

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


Main Theorem

Let $S_N \left({x}\right)$ denote the first $N$ terms of the Fourier series:

$(2): \quad S_N \left({x}\right) = \displaystyle \frac {a_0} 2 + \sum_{n \mathop = 1}^N \left({a_n \cos n x + b_n \sin n x}\right)$

where:

$(3): \quad \displaystyle a_n = \dfrac 1 \pi \int_\alpha^{\alpha + 2 \pi} f \left({x}\right) \cos n x \rd x$
$(4): \quad \displaystyle b_n = \dfrac 1 \pi \int_\alpha^{\alpha + 2 \pi} f \left({x}\right) \sin n x \rd x$


Substituting from $(3)$ and $(4)$ into $(2)$ and rearranging:

$S_N \left({x}\right) = \displaystyle \dfrac 1 \pi \int_\alpha^{\alpha + 2 \pi} f \left({u}\right) \left({\frac 1 2 + \sum_{n \mathop = 1}^N \left({\cos n x \cos n u + \sin n x \sin n u}\right)}\right) \rd u$

Now we have:

$\displaystyle\frac 1 2 + \sum_{n \mathop = 1}^N \left({\cos n x \cos n u + \sin n x \sin n u}\right) = \frac {\sin \left({\left({N + \frac 1 2}\right) \left({u - x}\right)}\right)} {2 \sin \left({\frac 1 2 \left({u - x}\right)}\right)}$



Hence:

$\displaystyle S_N \left({x}\right) = \int_\alpha^{\alpha + 2 \pi} \psi \left({u}\right) \frac {\sin \left({\left({N + \frac 1 2}\right) \left({u - x}\right)}\right)} {2 \sin \left({\frac 1 2 \left({u - x}\right)}\right)} \rd u$

where:

$\psi \left({u}\right) = \dfrac 1 \pi f \left({u}\right) \dfrac {\frac 1 2 \left({u - x}\right)} {\sin \frac 1 2 \left({u - x}\right)}$


We have that $f \left({u}\right)$ satisfies the Dirichlet conditions on $\left[{\alpha \,.\,.\, \alpha + 2 \pi}\right]$.



Hence $f$ is piecewise smooth on $\left[{\alpha \,.\,.\, \alpha + 2 \pi}\right]$.

That is, $f$ has right-hand derivative and left-hand derivative at all $x$ in $\left[{\alpha \,.\,.\, \alpha + 2 \pi}\right]$.

Thus at the point $u = x$, $f$ has right-hand derivative and left-hand derivative, and so does $\psi \left({u}\right)$.

So by Fourier's Theorem: Lemma 3:

$\displaystyle \lim_{n \mathop \to N} S_N \left({x}\right) = \frac \pi 2 \left({\psi \left({x^+}\right) + \psi \left({x^-}\right)}\right)$


Now:

$\psi \left({x^+}\right) = \displaystyle \frac 1 \pi f \left({x^+}\right) \lim_{u \mathop \to x} \dfrac {\frac 1 2 \left({u - x}\right)} {\sin \frac 1 2 \left({u - x}\right)} = \frac 1 \pi f \left({x^+}\right)$

and:

$\psi \left({x^-}\right) = \displaystyle \frac 1 \pi f \left({x^-}\right) \lim_{u \mathop \to x} \dfrac {\frac 1 2 \left({u - x}\right)} {\sin \frac 1 2 \left({u - x}\right)} = \frac 1 \pi f \left({x^-}\right)$

and so:

$\displaystyle \lim_{n \mathop \to N} S_N \left({x}\right) = \frac 1 2 \left({\lim_{x \mathop \to a^+} f \left({x}\right) + \lim_{x \mathop \to a^-} f \left({x}\right)}\right)$

$\blacksquare$


Also known as


Source of Name

This entry was named for Joseph Fourier.


Sources