Polynomials orthogonal on the interval
with unit weight
.
The
standardized Legendre polynomials
are defined by the
Rodrigues formula
and have the representation
The formulas most commonly used are:
The Legendre polynomials can be defined as the
coefficients in the expansion of the generating function
where the series on the right-hand side converges for
.
The first few standardized Legendre polynomials have the form
The Legendre polynomial of order
satisfies the differential equation
(
Legendre equation)
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
and satisfy the uniform and weighted estimates
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
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.