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--- math121a-f23:october_6_friday [2023/10/06 04:14]
pzhou created
+++ math121a-f23:october_6_friday [2026/02/21 14:41] (current)
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 ====== October 6 (Friday) ======
+Topics:
+  * Fourier inversion formula
+  * Fourier Series for periodic function
+  * Interpretation of complex vector space, hermitian inner product, orthonormal basis.
+===== Inversion Formula =====
+Suppose you started from $f(x)$, and did some hard work to get the Fourier transformation $F(p)$. Can you recover $f(x)$ from $F(p)$? Did you lose information when you throw away $f(x)$ and only keep $F(p)$?
+If $f(x)$ is continuous and absolutely integrable, we can recover $f(x)$ from $F(p)$ by
+$$ f(x) = \frac{1}{2\pi} \int_{p \in \R} F(p) e^{ipx} dp $$
+The proof of this theorem is beyond the scope of this class. You might be happy to just accept the formal 'rule' that
+$$ \frac{1}{2\pi} \int_\R e^{ipx - ipy} dp = \delta(x-y). $$
+and that
+$$ f(x) = \int \delta(x-y) f(y) dy $$
+We can try some example to see if it works.
+===== Fourer Series =====
+If the function $f(x)$ is a periodic function, of period $L$, meaning $f(x) = f(x+L)$, then we cannot do Fourier transform (why?), but instead, we need to do Fourier series.
+We are not going to use all $e^{ipx}$ for all $p$, but only those that satisfies $e^{ipx} = e^{ip(x+L)}$ have the same periodicity. Which mean $p$ needs to satisfy
+$$ p = (2\pi / L) n$$ for some integer $n$.
+So, we define
+$$ e_n(x) = e^{i(2\pi/L) n x}. $$
+We define the Fourer series coefficient as
+$$ c_n = \frac{1}{L} \int_{0}^L f(x) \overline{e_n(x)} dx $$
+Given these coefficient $c_n$, can we recover $f(x)$? Yes, under some smoothness condition of $f(x)$, we have
+$$ f(x) = \sum_{n \in \Z} c_n e_n(x). $$

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