Editing Generalized Fourier series

{{Short description|Decompositions of inner product spaces into orthonormal bases}}
{{Use American English|date = March 2019}}
{{tone|date=February 2024}}
A '''generalized Fourier series''' is the expansion of a [[square integrable]] function into a sum of square integrable [[orthogonal basis | orthogonal basis functions]]. The standard [[Fourier series]] uses an [[orthonormal basis]] of [[trigonometric functions]], and the series expansion is applied to periodic functions. In contrast, a generalized Fourier series uses any set of orthogonal basis functions and can apply to any [[Square-integrable function|square integrable function]].<ref>Herman p.82</ref><ref>Folland p.84</ref>

== Definition ==

Consider a set <math>\Phi = \{\phi_n:[a,b] \to \mathbb{C}\}_{n=0}^\infty</math> of [[square-integrable]] complex valued functions defined on the closed interval <math> [a,b] </math> that are pairwise [[orthogonal]] under the weighted [[inner product]]:

<math>\langle f, g \rangle_w = \int_a^b f(x) \overline{g(x)} w(x) dx,</math>

where <math>w(x)</math> is a [[weight function]] and <math>\overline g</math> is the [[complex conjugate]] of <math> g </math>. Then, the '''generalized Fourier series''' of a function <math> f </math> is:
<math display="block">f(x) = \sum_{n=0}^\infty c_n\phi_n(x),</math>where the coefficients are given by:
<math display="block">c_n = {\langle f, \phi_n \rangle_w\over \|\phi_n\|_w^2}.</math>

== Sturm-Liouville Problems ==
Given the space <math> L^2(a,b) </math> of square integrable functions defined on a given interval, one can find orthogonal bases by considering a class of boundary value problems on the interval <math> [a,b] </math> called [[Sturm-Liouville problems|regular Sturm-Liouville problems]]. These are defined as follows, 
<math display="block">
(rf')' + pf + \lambda wf = 0
</math>
<math display="block">
B_1(f) = B_2(f) = 0
</math>
where <math> r, r'</math> and <math> p </math> are real and continuous on <math> [a,b] </math> and <math> r > 0 </math> on <math> [a,b] </math>, <math> B_1 </math> and <math> B_2 </math> are [[self-adjoint]] boundary conditions, and <math> w </math> is a positive continuous functions on <math> [a,b] </math>.

Given a regular Sturm-Liouville problem as defined above, the set <math> \{\phi_n\}_{1}^{\infty} </math> of [[eigenfunctions]] corresponding to the distinct [[eigenvalue]] solutions to the problem form an orthogonal basis for <math> L^2(a,b) </math> with respect to the weighted inner product <math> \langle\cdot,\cdot\rangle_w </math>.<ref>Folland p.89</ref> We also have that for a function <math> f \in L^2(a,b) </math> that satisfies the boundary conditions of this Sturm-Liouville problem, the series <math> \sum_{n=1}^{\infty} \langle f,\phi_n \rangle \phi_n </math> [[converges uniformly]] to <math> f </math>.<ref>Folland p.90</ref>

== Examples ==
=== Fourier–Legendre series ===
A function <math>f(x)</math> defined on the entire number line is called [[periodic function|periodic]] with period <math>T</math> if a number <math>T>0</math> exists such that, for any real number <math>x</math>, the equality <math>f(x+T)=f(x)</math> holds.

If a function is periodic with period <math>T</math>, then it is also periodic with periods <math>2T</math>, <math>3T</math>, and so on. Usually, the period of a function is understood as the smallest such number <math>T</math>. However, for some functions, arbitrarily small values of <math>T</math> exist.

The sequence of functions <math>1, \cos(x), \sin(x), \cos(2x), \sin(2x),..., \cos(nx), \sin(nx),...</math> is known as the trigonometric system. Any [[linear combination]] of functions of a trigonometric system, including an infinite combination (that is, a converging [[infinite series]]), is a periodic function with a period of 2π.

On any segment of length 2π (such as the segments [−π,π] and [0,2π]) the trigonometric system is an [[orthogonal system]]. This means that for any two functions of the trigonometric system, the integral of their product over a segment of length 2π is equal to zero. This integral can be treated as a [[scalar product]] in the space of functions that are integrable on a given segment of length 2π.

Let the function <math>f(x)</math> be defined on the segment [−π, π]. Given appropriate smoothness and differentiability conditions, <math>f(x)</math> may be represented on this segment as a linear combination of functions of the trigonometric system, also referred to as the ''expansion'' of the function <math>f(x)</math> into a trigonometric Fourier series.

The [[Legendre polynomials]] <math>P_n(x)</math> are solutions to the [[Sturm–Liouville theory|Sturm–Liouville]] eigenvalue problem

: <math> \left((1-x^2)P_n'(x)\right)'+n(n+1)P_n(x)=0.</math>

As a consequence of Sturm-Liouville theory, these polynomials are orthogonal [[eigenfunction]]s with respect to the [[inner product]] with unit weight. This can be written as a generalized Fourier series (known in this case as a Fourier–Legendre series) involving the Legendre polynomials, so that

:<math>f(x) \sim \sum_{n=0}^\infty c_n P_n(x),</math>
:<math>c_n = {\langle f, P_n \rangle_w\over \|P_n\|_w^2}</math>

As an example, the Fourier–Legendre series may be calculated for <math>f(x)=\cos x</math> over <math>[-1, 1]</math>. Then

:<math>
\begin{align}
c_0 & = {\int_{-1}^1 \cos{x}\,dx \over \int_{-1}^1 (1)^2 \,dx}  = \sin{1} \\
c_1 &  = {\int_{-1}^1 x \cos{x}\,dx \over \int_{-1}^1 x^2 \, dx} = {0 \over 2/3 } =0 \\
c_2 & =  {\int_{-1}^1 {3x^2 - 1 \over 2} \cos{x} \, dx \over \int_{-1}^1 {9x^4-6x^2+1 \over 4} \, dx} = {6 \cos{1} - 4\sin{1} \over 2/5 }
\end{align}
</math>

and a truncated series involving only these terms would be

:<math>\begin{align}c_2P_2(x)+c_1P_1(x)+c_0P_0(x)&= {5 \over 2} (6 \cos{1} - 4\sin{1})\left({3x^2 - 1 \over 2}\right) + \sin1\\
&= \left({45 \over 2} \cos{1} - 15 \sin{1}\right)x^2+6 \sin{1} - {15 \over 2}\cos{1}\end{align}</math>

which differs from <math>\cos x</math> by approximately 0.003. In computational applications it may be advantageous to use such Fourier–Legendre series rather than Fourier series since the basis functions for the series expansion are all polynomials and hence the integrals and thus the coefficients may be easier to calculate.

== Coefficient theorems ==
Some theorems on the series' coefficients <math>c_n</math> include:

=== [[Bessel's inequality]] ===
'''Bessel's inequality''' is a statement about the coefficients of an element <math>x</math> in a [[Hilbert space]] with respect to an [[orthonormal]] [[sequence]]. The inequality was derived by [[Frederic Bessel|F.W. Bessel]] in 1828:<ref>{{Cite web|url=https://www.encyclopediaofmath.org/index.php/Bessel_inequality|title = Bessel inequality - Encyclopedia of Mathematics}}</ref>
:<math>\sum_{n=0}^\infty |c_n|^2\leq\int_a^b|f(x)|^2w(x)\,dx.</math>

=== [[Parseval's theorem]] ===
'''Parseval's theorem''' usually refers to the result that the [[Fourier transform]] is [[Unitary operator|unitary]]; loosely, that the sum (or integral) of the square of a function is equal to the sum (or integral) of the square of its transform.<ref>Parseval des Chênes, Marc-Antoine Mémoire sur les séries et sur l'intégration complète d'une équation aux différences partielles linéaire du second ordre, à coefficients constants" presented before the Académie des Sciences (Paris) on 5 April 1799. This article was published in ''Mémoires présentés à l’Institut des Sciences, Lettres et Arts, par divers savants, et lus dans ses assemblées. Sciences, mathématiques et physiques. (Savants étrangers.)'', vol. 1, pages 638–648 (1806).</ref>

If Φ is a complete basis, then:
:<math> \sum_{n=0}^\infty |c_n|^2 = \int_a^b |f(x)|^2w(x)\, dx.</math>

== See also ==
* [[Banach space]]
* [[Eigenfunctions]]
* [[Fractional Fourier transform]]
* [[Function space]]
* [[Hilbert space]]
* [[Least-squares spectral analysis]]
* [[Orthogonal function]]
* [[Orthogonality]]
* [[Topological vector space]]
* [[Vector space]]

== References ==
{{reflist}}
*[https://mathworld.wolfram.com/GeneralizedFourierSeries.html Generalized Fourier Series] at ''[[MathWorld]]''
*{{cite book 
    | last=Herman 
    | first= Russell 
    | title= An Introductions to Fourier and Complex Analysis with Applications to the Spectral Analysis of Signals 
    | date=2016
    | pages=73–112
    | url = https://people.uncw.edu/hermanr/mat367/fcabook/FCA_Main.pdf
}}
*{{cite book 
    | last=Folland 
    | first=Gerald B. 
    | title=Fourier Analysis and Its Applications 
    | date=1992 
    | author-link=Gerald Folland 
    | publisher=Wadsworth & Brooks/Cole Advanced Books & Software 
    | location=Pacific Grove, California
    | pages=62–97
    | url= https://www-elec.inaoep.mx/~rogerio/Tres/FourierAnalysisUno.pdf
}}

[[Category:Fourier analysis]]