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Figure 21.5.
A function that is not of bounded
variation.
With these preliminaries out of the way, we can get to the main theorems about
Fourier series.
21.5.6
Theorem.
(1) If f(t) is a periodic function of period 2p, which is absolutely integrable and
of bounded variation on [-p,p], then its Fourier series (21.10) converges to
()
+
()
2
+
-
ft
ft
for each t. (f(t
±
) refers to one-sided limits of f at t.) If the function is also continuous,
then the Fourier series converges to f(t) for each t.
(2) If two continuous periodic functions f and g of period 2p have the same
Fourier coefficients, then f = g.
Proof.
See [Apos58] and [Seel66].
Theorem 21.5.6 deals with how Fourier series converge
pointwise
to the periodic
functions that defined them. However, we can also ask about convergence in the L
2
norm. The other basic question is then answered by the following theorem:
21.5.7 Theorem.
Let f be an arbitrary (not necessarily periodic) function in
L
2
([-p,p]). Then the Fourier series (21.10) converge in the L
2
metric and is equal to f.
(Recall that equality here means that the integral of the difference is zero.)
Proof.
See [Nata61]. The fact that f is not periodic is not a problem. Since we are
only interested in what happens in the interval [-p,p] we can forget whatever defini-
tion it might have outside that interval and, as Example 21.5.2 showed, we can make
it into a periodic function on
R
of period 2p with the formula f(t + 2kp) = f(t), where
k = 0 , ±1, ±2,..., and t Œ [-p,p] .
Note that Theorem 21.5.7 does
not
say that the Fourier series of f converges point-
wise to f. It may seem strange that one would be interested in non-pointwise conver-
gence, but in fact L
2
convergence is a useful type of convergence.
Another variant of this theorem is
