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| -rw-r--r-- | ClassJuly5.page | 6 |
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diff --git a/ClassJuly5.page b/ClassJuly5.page index 7e19aff..f6b879d 100644 --- a/ClassJuly5.page +++ b/ClassJuly5.page @@ -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 -$$ \lim_{N \to \infty} \sqrt{\int_0^{2\pi} \left| \sum_{n = - N}^N c_n e^{in\theta} - f(\theta) \right|^2} = 0 $$
\ No newline at end of file +$$ \lim_{N \to \infty} \sqrt{\int_0^{2\pi} \left| f(\theta) - \sum_{n = - N}^N c_n e^{in\theta} \right|^2} = 0 $$ + +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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