4.4 Spherical Hannonics
205
equal to m - 1. Since the polynomial q can be expressed as a linear combination
of the Legendre polynomials of order less than or equal to m - 1, all of whieh are
orthogonal to Pm,
i
b
f(x)dx = i
b
q(x)Pm(x)dx + i
b
r(x)dx = i
b
r(x)dx .
(4.61)
Also,
m
m
m
LAj/(Xj) = LAjq(xj)Pm(Xj) + LAjr(xj)
j = l
j=1
m
j=l
= LAjr(xj)
j=l
= i
b
r(x)dx ,
(4.62)
where the second equality holds because the x j are the zeros of Pm, and the third
equality is obtained because r is a polynomial of order less than or equal to m - 1.
It follows from (4.61) and (4.62) that
and that m-point Gaussian quadrature is exact for polynomials up to order 2m - 1.
For m-point Gaussian quadrature over the domain [-1, 1],
Formulae for the x j are not known in closed form and must be computed numerically. This can be done using Newton's method (Dahlquist and Björck 1974) with
first guesses for the m zeros of Pm(X) given by the set of points
4j - 1 )
Xj= -cos [( 2m+1
rrJ
"2 for i s j s m.
Avoiding Aliasing Error
The product oftwo or more truncated spectral harmonie expansions contains highorder Fourier modes in A and high-order functions in J1, that are not present in
the original truncation. When using the transform method it is important to retain
enough zonal wave numbers in the Fourier transforms and enough meridional grid
points in the Gaussian quadrature to ensure that the transform procedure does not
generate errors in any of the modes retained in the original truncated expansions.