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--- math121a-f23:october_6_friday [2023/10/05 21:14]
pzhou created
+++ math121a-f23:october_6_friday [2023/10/05 22:45] (current)
pzhou [October 6 (Friday)]
@@ Line 1: / Line 1: @@
 ====== 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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