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| + | ==== Fourier convergence theorem ==== | ||
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| + | We will first start by reviewing Lecture 19 and our introduction of the Fourier convergence theorem. This involves **Theorem 12.5** in the notes. In particular, the theorem states that if $f$ is a $2\pi$-periodic function with $f$ and $f'$ continuous on the interval $(-\pi, \pi)$, then the Fourier series of $f$ at $x$ converges to the average of the left- and right-limits. Thus | ||
| + | $$ | ||
| + | \frac{1}{2}[f(x_-) + f(x_+)] | ||
| + | $$ | ||
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| + | We'll draw some pictures of how to visualise the theorem. We will also discuss again this notion of pointwise convergence vs. uniform convergence and the notion of the Gibbs' Phenomenon. | ||
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| + | ==== Fourier series on any interval ==== | ||
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| + | Next, we want to simply note that the Fourier series you've derived for $2\pi$-periodic functions on $[-\pi, \pi]$ can be easily extended to functions defined on $[-L, L]$. The truth is that we should really just have done the derivation like this from the get-go! This leads to: | ||
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| + | **Theorem 12.7:** (Fourier coefficients for $2L$-periodic function) Let $f$ be a periodic function with period $2L$. Then | ||
| + | $$ | ||
| + | f(x) \sim \frac{a_0}{2} + \sum_{n=1}^\infty \left[ a_n \cos\left(\frac{n\pi x}{L}\right) + b_n \sin\left(\frac{n\pi x}{L}\right)\right] | ||
| + | $$ | ||
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