Definition:Fourier Series/Range 2 Pi

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Definition

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

Let $f: \R \to \R$ be a function such that $\ds \int_\alpha^{\alpha + 2 \pi} \map f x \rd x$ converges absolutely.


Let:

\(\ds a_n\) \(=\) \(\ds \dfrac 1 \pi \int_\alpha^{\alpha + 2 \pi} \map f x \cos n x \rd x\)
\(\ds b_n\) \(=\) \(\ds \dfrac 1 \pi \int_\alpha^{\alpha + 2 \pi} \map f x \sin n x \rd x\)


Then:

$\dfrac {a_0} 2 + \ds \sum_{n \mathop = 1}^\infty \paren {a_n \cos n x + b_n \sin n x}$

is called the Fourier Series for $f$.


Fourier Coefficient

The constants:

$a_0, a_1, a_2, \ldots, a_n, \ldots; b_1, b_2, \ldots, b_n, \ldots$

are the Fourier coefficients of $f$.


Fourier Series on General Range

The range of the Fourier series may be extended to any general real interval:


Formulation 1

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

Let $\lambda \in \R_{>0}$ be a strictly positive real number.

Let $f: \R \to \R$ be a function such that $\ds \int_\alpha^{\alpha + 2 \lambda} \map f x \rd x$ converges absolutely.


Let:

\(\ds a_n\) \(=\) \(\ds \dfrac 1 \lambda \int_\alpha^{\alpha + 2 \lambda} \map f x \cos \frac {n \pi x} \lambda \rd x\)
\(\ds b_n\) \(=\) \(\ds \dfrac 1 \lambda \int_\alpha^{\alpha + 2 \lambda} \map f x \sin \frac {n \pi x} \lambda \rd x\)


Then:

$\ds \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty \paren {a_n \cos \frac {n \pi x} \lambda + b_n \sin \frac {n \pi x} \lambda}$

is the Fourier Series for $f$.


Formulation 2

Let $a, b \in \R$ be real numbers.

Let $f: \R \to \R$ be a function such that $\ds \int_a^b \map f x \rd x$ converges absolutely.


Let:

\(\ds A_m\) \(=\) \(\ds \dfrac 2 {b - a} \int_a^b \map f x \cos \frac {2 m \pi \paren {x - a} } {b - a} \rd x\)
\(\ds B_m\) \(=\) \(\ds \dfrac 2 {b - a} \int_a^b \map f x \sin \frac {2 m \pi \paren {x - a} } {b - a} \rd x\)


Then:

$\ds \frac {A_0} 2 + \sum_{m \mathop = 1}^\infty \paren {A_m \cos \frac {2 m \pi \paren {x - a} } {b - a} + B_m \sin \frac {2 m \pi \paren {x - a} } {b - a} }$

is the Fourier Series for $f$.


Also known as

The Fourier series, as defined on the general open interval $\openint \alpha {\alpha + 2 \lambda}$, can often be seen referred to as the whole-range Fourier series.

This is in order to distinguish it from the $2$ varieties of half-range Fourier series, which are applied to the open interval $\openint 0 \lambda$.


Also defined as

The form of the Fourier series given here is more general than that usually given.

The usual form is one of the cases where $\alpha = 0$ or $\alpha = -\pi$, thus giving a range of integration of either $\openint 0 {2 \pi}$ or $\openint {-\pi} \pi$.

The actual range may often be chosen for convenience of analysis.


Also see

  • Results about Fourier series can be found here.


Source of Name

This entry was named for Joseph Fourier.


Historical Note

Despite the fact that the Fourier series bears the name of Joseph Fourier, they were first studied by Leonhard Paul Euler.

Fourier himself made considerable use of this series during the course of his analysis of the behaviour of heat.

The first person to feel the need for a careful study of its convergence was Augustin Louis Cauchy.

In $1829$, Johann Peter Gustav Lejeune Dirichlet gave the first satisfactory proof about the sums of Fourier series for certain types of function.

The criteria set by Dirichlet were extended and generalized by Riemann in his $1854$ paper Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe.


Sources

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