674
H The Fourier Decomposition
where the g(k) are the coefficients of the expansion. For the convenience which
follows, the simpler functions φ(k, x) are usually taken as orthogonal for different k:
φ
∗ (k
, x)φ(k, x)dx = δ(k
− k) ,
(H.12)
where δ(k − k) is the Dirac delta, defined by the property
δ(k
− k)h(k
)dk
= h(k)
(H.13)
for any piecewise continuous function h(k).
One can then solve for the coefficients g(k) as
g(k) =
φ
∗ (k, x)f (x)dx .
(H.14)
So the g(k) are the generalized Fourier transform of the original f (x). If any
function f (x) defined over x within the given boundaries can be so expanded,
then we say the expansion functions φ(k, x) are complete. Completeness can be
expressed as
φ(k, x
)φ
∗ (k, x)dk = δ(x
− x)
(H.15)
since this relation can be placed in
f (x) =
δ(x
− x)f (x
)dx
(H.16)
to generate the expansion of f (x).
Each of the above relations can be re-written as a discrete sum to represent the
corresponding relations for a generalized Fourier series. For some useful cases, the
complete set of expansion functions contains both discrete and continuous function
labels k.
Many of the linear differential equations in physics are of the Strum-Liouville
type. In these cases, the expansion functions φ(k, x) can be taken as eigenfunctions
of the corresponding Strum-Liouville equation:
d
dx
p(x)
d
dx
+ q(x)
φ(k, x) = −λ(k)w(x)φ(k, x) .
(H.17)
With fixed finite boundary conditions on the solutions, only certain ‘eigenvalues’
λ(k) give a solution. Each possible eigenvalue is labeled by ‘k’. Then, the
arbitrary solution for the same boundary conditions is a linear combination of the
eigenfunctions for the various k values.