Legendre polynomials
spherical polynomials
Polynomials orthogonal on the interval with unit weight
. The standardized Legendre polynomials are defined by the Rodrigues formula
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and have the representation
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The formulas most commonly used are:
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The Legendre polynomials can be defined as the coefficients in the expansion of the generating function
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where the series on the right-hand side converges for .
The first few standardized Legendre polynomials have the form
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The Legendre polynomial of order satisfies the differential equation (Legendre equation)
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which occurs in the solution of the Laplace equation in spherical coordinates by the method of separation of variables. The orthogonal Legendre polynomials have the form
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and satisfy the uniform and weighted estimates
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Fourier series in the Legendre polynomials inside the interval are analogous to trigonometric Fourier series (cf. also Fourier series in orthogonal polynomials); there is a theorem about the equiconvergence of these two series, which implies that the Fourier–Legendre series of a function
at a point
converges if and only if the trigonometric Fourier series of the function
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converges at the point . In a neighbourhood of the end points the situation is different, since the sequence
increases with speed
. If
is continuous on
and satisfies a Lipschitz condition of order
, then the Fourier–Legendre series converges to
uniformly on the whole interval
. If
, then this series generally diverges at the points
.
These polynomials were introduced by A.M. Legendre [1].
See also the references to Orthogonal polynomials.
References
[1] | A.M. Legendre, Mém. Math. Phys. présentés à l'Acad. Sci. par divers savants , 10 (1785) pp. 411–434 |
[2] | E.W. Hobson, "The theory of spherical and ellipsoidal harmonics" , Cambridge Univ. Press (1931) |
Comments
Legendre polynomials belong to the families of Gegenbauer polynomials; Jacobi polynomials and classical orthogonal polynomials. They can be written as hypergeometric functions (cf. Hypergeometric function). Their group-theoretic interpretation as zonal spherical functions on the two-dimensional sphere serves as a prototype, both from the historical and the didactical point of view. A noteworthy consequence of this interpretation is the addition formula for Legendre polynomials.
Legendre polynomials. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Legendre_polynomials&oldid=17589