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516 CHAPTER 15 Special Functions and Eigenfunction Expansions

f · ϕ n f · ϕ n
c n = = .
ϕ n · ϕ n ϕ n 2
We will use some facts from linear algebra to derive an important property of these
coefficients. Suppose we write a linear combination

N

k n ϕ n (x),
n=1
N
with the k n ’s any numbers. We ask: how should we choose the k n ’s so that n=1 k n ϕ n (x) best
approximates f (x) for a < x < b? This question is well posed because we have a notion of
distance between functions. The problem is to choose the k n ’s to minimize the distance between
N
f (x) and k n ϕ n (x):
n=1

N

f − k n ϕ n (x) .

n=1
We know the answer to this question if we think in the context of orthogonal projections in
a vector space. Let PC[a,b] be the vector space of piecewise continuous functions defined on
[a,b]. This is a vector space because sums and scalar multiples of piecewise continuous functions
are also piecewise continuous, and the zero function is piecewise continuous. The eigenfunctions
ϕ 1 (x),ϕ 2 (x),··· ,ϕ N (x)
span a subspace S of PC[a,b], and in fact form an orthogonal basis for this subspace. The
linear combination of these eigenfunctions having minimum distance from f is the orthogo-
nal projection f S of f onto this subspace, and we know from Sections 6.6 and 6.7 that this
projection is
N
f · ϕ n

f S (x) = ϕ n (x).
ϕ n · ϕ n
n=1
This is exactly the Nth partial sum of the eigenfunction expansion of f .
We may therefore think of the coefficients in the eigenfunction expansion of f as those
for which the Nth partial sum of the expansion is the best approximation to f (x) as a linear
combinations of the first N eigenfunctions.
There is also a Bessel inequality and a Parseval theorem for general eigenfunction expansion
coefficients. First, let
ϕ n
n = ,
ϕ n
which we can also write as
ϕ n
n = √ .
ϕ n · ϕ n
This divides each eigenfunction by its length, resulting in an eigenfunction of length 1:
2
n = n · n = 1.
We say that the eigenfunctions have been normalized. Since nonzero constant multiples
of eigenfunctions are eigenfunctions, we can expand f (x) in a series of these normalized
eigenfunctions
∞

c n n (x),
n=1

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October 14, 2010 15:20 THM/NEIL Page-516 27410_15_ch15_p505-562