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SECTION 10.3 WHEN DOES A FOURIER SERIES CONVERGE?
THE FOURIER CONVERGENCE THEOREM.
Suppose that
f
is originally deﬁned
for

L
≤
x < L
and the deﬁnition is then extended so that
f
is periodic with period 2
L
.
Suppose also that then
f
and
f
0
are piecewise continuous on the interval

L
≤
x
≤
L
. Then
the Fourier series whose coeﬃcients are given by the EulerFourier formulas converges to
f
(
x
)
at points where
f
is continuous and to
f
(
x
+) +
f
(
x

)
2
at points where
f
is discontinuous.
Piecewise continuous
means that
f
is continuous everywhere except for ﬁnitely many
points
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 Spring '10
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