It is important to establish simple criteria which determine when a Fourier series
converges.
In this section we will develop conditions on f(x) that enable us to determine the sum
of the Fourier series.
One quite useful method to analyse the convergence properties is to express the partial
sums of a Fourier series as integrals. Riemann and Fejer have since provided other ways
of summing Fourier series.
In this section we limit the study of convergence to functions that are piecewise smooth
on a given interval.
Definition 5 (Piecewise smooth function)
A function f is piecewise smooth on an interval if both f and f' are piecewise
continuous on the interval.
Theorem 2
Suppose that
f is piecewise smooth and periodic.
Then the series (
1) with coefficients (
2) converges to
 1.
 f(x) if x is a point of continuity.
 2.

if x is a point of discontinuity.
This means that, at each x between L and L, the Fourier series converges
to the average of the left and the right limits of f(x) at x. If fis continuous at x, then the left and the right limits are both equal to f(x),
and the Fourier series converges to f(x) itself. If f has a jump discontinuity at xthen the Fourier series converges to the point midway in the gap at this point.
Theorem 3 (Bessel's inequality)
Suppose that
f is integrable on the interval [0,
T]. Let
a_{n},
b_{n},
c_{n} be the Fourier
coefficients of
f. Then

(8) 
Theorem 4 (Riemann lemma)
Let
f be integrable and
a_{n} and
b_{n} be the Fourier coefficients of
f. Then
which means
Theorem 5 (Parseval's identity)

(9) 
if
a_{n} and
b_{n} are the Fourier coefficients corresponding to
f(
x) and if
f(
x)satisfies the Dirichlet conditions.
19981027 (Marian Prutscher)
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