Definition:Fourier Series
Definition
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 $f$.
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 Range of $2 \pi$
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 $f$.
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
- Coefficients of Cosine Terms in Convergent Trigonometric Series
- Coefficients of Sine Terms in Convergent Trigonometric Series
- 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.