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15.1 Eigenfunction Expansions 517

where
f · n
c n = = f · n .
n · n
Now,

1if n = m,
n · m =
0if n = m.
Then,
2
N N N

0 ≤ f − c n n = f − c n n · f − c m m

n=1 n=1 n=1

N N N

= f · f − 2 c n f · n + c n c m n · m
n=1 n=1 m=1
N N

2
= f · f − 2 c + c 2
n n
n=1 n=1
N

2
= f · f − c .
n
n=1
Therefore,
N
2
c ≤ f · f.
n
n=1
This is Bessel’s inequality for this general setting. It says that the sum of the first N squares
of the generalized Fourier coefficients of f is less than or equal to the length of f . Since
N can be any positive number, this implies that
∞
2 2
c ≤ f .
n
n=1
The series of squares of these coefficients converges, and the sum is bounded by the square
of the length of f .
∞

If f is continuous on [a,b], then f (x) = c n n (x) on (a,b) and in this case
n=1
Bessel’s inequality is an equality:
∞

2
2
c = f ,
n
n=1
or, equivalently,
∞

2
c = f · f.
n
n=1
This is Parseval’s theorem.
Both of these results were seen previously in the special case of Fourier series in Theorems
13.6 and 13.7.

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October 14, 2010 15:20 THM/NEIL Page-517 27410_15_ch15_p505-562