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Fourier-Legendre Series


Because the Legendre polynomials form a complete orthogonal system over the interval [-1,1] with respect to the weighting function w(x)=1, any function f(x) may be expanded in terms of them as

 f(x)=sum_(n=0)^inftya_nP_n(x).
(1)

To obtain the coefficients a_n in the expansion, multiply both sides by P_m(x) and integrate

 int_(-1)^1P_m(x)f(x)dx=sum_(n=0)^inftya_nint_(-1)^1P_n(x)P_m(x)dx.
(2)

But the Legendre polynomials obey the orthogonality relationship

 int_(-1)^1P_n(x)P_m(x)dx=2/(2m+1)delta_(mn),
(3)

where delta_(mn) is the Kronecker delta, so

int_(-1)^1P_m(x)f(x)dx=sum_(n=0)^(infty)a_n2/(2m+1)delta_(mn)
(4)
=2/(2m+1)a_m
(5)

and

 a_m=(2m+1)/2int_(-1)^1P_m(x)f(x)dx.
(6)

For example, for f(x)=sin(pix), the first few terms of the Fourier-Legendre series are

 f(x)=3/piP_1(x)+(7(pi^2-15))/(pi^3)P_3(x)+(11(pi^4-105pi^2+945))/(pi^5)P_5(x)+....
(7)

See also

Fourier-Bessel Series, Fourier Series, Generalized Fourier Series, Jackson's Theorem, Laplace Series, Legendre Polynomial, Picone's Theorem

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References

Kaplan, W. "Fourier-Legendre Series." §7.14 in Advanced Calculus, 4th ed. Reading, MA: Addison-Wesley, pp. 508-512, 1992.

Referenced on Wolfram|Alpha

Fourier-Legendre Series

Cite this as:

Weisstein, Eric W. "Fourier-Legendre Series." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Fourier-LegendreSeries.html

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