i
21t i1t
'Yn,m = 41t~
g(a,A) PnJ~(cosa) e- imA sina da dA
n ~ 0, -n::;; m ::;; n (7)
moo
where em = 2 - o(m,O), 0 being the delta function. Since g is real, 'Yn,-m = 'Y:,m. The
integers nand m are, respectively, the degree and order of the harmonic coefficient, 'Yn,m'
The P n,m are called associated Legendre functions of the first kind, defined, for the present
purpose, for non-negative integers n,m with m ::;; n and for 0 ::;; S ::;; 1t. Furthermore, they
are assumed to be normalized such that
L' Pj,m(cos9) Pn,m(cos9) sin9 d9 = I~;
j¢n
j=n
(again, for simplicity, the conventional over-bar notation is omitted).
(8)
The relationships between the Legendre and Fourier spectra were derived by Jekeli
(l995):
00
G
"'"
nJrnj
k,m = ~ 'Yn,m Pk
n=l~
- ~ "'" G an J~
'Yn,m - 2e ~ k,m -k
m k=-oo
(9a,b)
where
p~,m = ~f Pn,m(9) e,i2k (lOa,b)
It is noted that because P nm(a) and P nm(a)sina are defined on [0,1t], their spectra (lOa,b)
contain an infinity of components. Also, the orthogon,lity ~f the associated Legendre
functions implies that the infinite sequences {p~,m} and t ~km~ are orthogonal for n ¢ j.
Indeed, substituting the series for P nm(S) and P nm(a)sinS in terms of their Fourier spectra
into (8) yields
00
L pt m a~km = 2!m o(j,n)
k=-oo
(11)
Other useful formulas relate (lOa,b) to the corresponding discrete spectra of the sequence
of values of the functions P n,m (a) and P n,m (a )sina sampled on a grid,
{as = s AS : s = 0, ... , K-l; AS = 1t/K}. Specializing (5) to a single dimension, one has
immediately:
00
P o kn,m = "'" n,m
~ PuK+k ;
u=-oo
00
~k,m = "'" ~
wK+k
w--(12a,b)
The coefficients p~,m and ~,m are computed using the DFf of Pn,m(S) and Pn,m(S)sinS
(see Jekeli, 1995).
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