A bit about L2 convergence

1 files changed, 4 insertions, 2 deletions

diff --git a/ClassJuly5.page b/ClassJuly5.page
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@@ -73,6 +73,8 @@ In fact, every function of the kind described above does have a Fourier sine exp
 
 How do we know that the Fourier series of a square wave or sawtooth function converges?
 
-The answer to this question depends greatly on the type of convergence desired.  Aside from the convergence we already proved, the next easiest type of convergence is $L^2$ or root-mean-square convergence.  The formal statement is that
+The answer to this question depends greatly on the type of convergence desired.  Aside from the convergence we already proved, the next easiest type of convergence is $L^2$ convergence.  The formal statement is that
 
-$$ \lim_{N \to \infty} \sqrt{\int_0^{2\pi} \left| \sum_{n = - N}^N c_n e^{in\theta} - f(\theta) \right|^2} = 0 $$ 
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+$$ \lim_{N \to \infty} \sqrt{\int_0^{2\pi} \left| f(\theta) - \sum_{n = - N}^N c_n e^{in\theta} \right|^2} = 0 $$ 
+
+In other words, as we let $N$ go to infinity, the root mean square error of our approximation gets arbitrarily small.  
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