Editing Generalized Fourier series (section)

=== 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.