generalized Fourier series

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

$\backslash langle\; f,\; g\; \backslash rangle\_w\; =\; \backslash int\_a^b\; f(x)\; \backslash overline\{g(x)\}\; w(x)\; dx,$

where $w(x)$ is a weight function and $\backslash overline\; g$ is the complex conjugate of $g$. Then, the generalized Fourier series of a function $f$ is:

$$f(x)\; =\; \backslash sum\_\{n=0\}^\backslash infty\; c\_n\backslash phi\_n(x),$$where the coefficients are given by:

$$c\_n\; =\; \{\backslash langle\; f,\; \backslash phi\_n\; \backslash rangle\_w\backslash over\; \backslash |\backslash phi\_n\backslash |\_w^2\}.$$

Given the space $L^2(a,b)$ 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 $[a,b]$ called regular Sturm-Liouville problems. These are defined as follows,

(rf')' + pf + \lambda wf = 0

B_1(f) = B_2(f) = 0

where $r,\; r\text{'}$ and $p$ are real and continuous on $[a,b]$ and $r\; >\; 0$ on $[a,b]$, $B\_1$ and $B\_2$ are self-adjoint boundary conditions, and $w$ is a positive continuous functions on $[a,b]$.

Given a regular Sturm-Liouville problem as defined above, the set $\backslash \{\backslash phi\_n\backslash \}\_\{1\}^\{\backslash infty\}$ of eigenfunctions corresponding to the distinct eigenvalue solutions to the problem form an orthogonal basis for $L^2(a,b)$ with respect to the weighted inner product $\backslash langle\backslash cdot,\backslash cdot\backslash rangle\_w$.Folland p.89 We also have that for a function $f\; \backslash in\; L^2(a,b)$ that satisfies the boundary conditions of this Sturm-Liouville problem, the series $\backslash sum\_\{n=1\}^\{\backslash infty\}\; \backslash langle\; f,\backslash phi\_n\; \backslash rangle\; \backslash phi\_n$ converges uniformly to $f$.Folland p.90

A function $f(x)$ defined on the entire number line is called periodic with period $T$ if a number $T>0$ exists such that, for any real number $x$, the equality $f(x+T)=f(x)$ holds.

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

The sequence of functions $1,\; \backslash cos(x),\; \backslash sin(x),\; \backslash cos(2x),\; \backslash sin(2x),...,\; \backslash cos(nx),\; \backslash sin(nx),...$ 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 $f(x)$ be defined on the segment [−π, π]. Given appropriate smoothness and differentiability conditions, $f(x)$ 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 $f(x)$ into a trigonometric Fourier series.

The Legendre polynomials $P\_n(x)$ are solutions to the Sturm–Liouville eigenvalue problem

: $\backslash left((1-x^2)P\_n\text{'}(x)\backslash right)\text{'}+n(n+1)P\_n(x)=0.$

As a consequence of Sturm-Liouville theory, these polynomials are orthogonal eigenfunctions 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

:$f(x)\; \backslash sim\; \backslash sum\_\{n=0\}^\backslash infty\; c\_n\; P\_n(x),$

:$c\_n\; =\; \{\backslash langle\; f,\; P\_n\; \backslash rangle\_w\backslash over\; \backslash |P\_n\backslash |\_w^2\}$

As an example, the Fourier–Legendre series may be calculated for $f(x)=\backslash cos\; x$ over $[-1,\; 1]$. Then

:$$

\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}

and a truncated series involving only these terms would be

:

&= \left({45 \over 2} \cos{1} - 15 \sin{1}\right)x^2+6 \sin{1} - {15 \over 2}\cos{1}\end{align}

which differs from $\backslash cos\; x$ 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.

Some theorems on the series' coefficients $c\_n$ include:

Bessel's inequality is a statement about the coefficients of an element $x$ in a Hilbert space with respect to an orthonormal sequence. The inequality was derived by F.W. Bessel in 1828:{{Cite web|url=https://www.encyclopediaofmath.org/index.php/Bessel_inequality|title = Bessel inequality - Encyclopedia of Mathematics}}

:$\backslash sum\_\{n=0\}^\backslash infty\; |c\_n|^2\backslash leq\backslash int\_a^b|f(x)|^2w(x)\backslash ,dx.$

Parseval's theorem usually refers to the result that the Fourier transform is 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.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).

If Φ is a complete basis, then:

:$\backslash sum\_\{n=0\}^\backslash infty\; |c\_n|^2\; =\; \backslash int\_a^b\; |f(x)|^2w(x)\backslash ,\; dx.$

• Banach space
• Eigenfunctions
• Fractional Fourier transform
• Function space
• Hilbert space
• Least-squares spectral analysis
• Orthogonal function
• Orthogonality
• Topological vector space
• Vector space

{{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