This section provides a brief introduction to Fourier Series. (Note: Zach Weiner's explanation isn't entirely accurate.) We work with the Hilbert space $L^2([0,L])$ of square-integrable functions on the interval $[0,L]$, with inner product $$ \langle {\bf f} | {\bf g} \rangle = \int_0^L \overline{f(t)} g(t) dt.$$ The functions $$ \xi_n(t) = \sin\left ( \frac{n \pi t}{L} \right )$$ are orthogonal, with $$ \langle {\bf \xi_m} | {\bf \xi_n} \rangle = \begin{cases} L/2 & n=m \cr 0 & n \ne m. \end{cases} $$ So if a vector $| {\bf f} \rangle$ decomposes as $\sum_n f_n | \xi_n \rangle$, then the coefficients are $f_n = \langle {\bf \xi_n} | {\bf f} \rangle/ \langle {\bf \xi_n} | {\bf \xi_n} \rangle = \frac{2}{L} \langle {\bf \xi_n} | {\bf f} \rangle.$ In terms of functions and integrals, this becomes: \begin{eqnarray*} f(t) & = & \sum_{n = 1}^\infty f_n \sin \left ( \frac{n \pi t}{L} \right ) \cr f_n & = & \frac{2}{L} \int_0^L f(t) \sin \left ( \frac{n \pi t}{L} \right ) dt. \end{eqnarray*} This is laid out in the first video, where we also work an example in detail.

Fourier series is particularly important in physics, since sine functions are normal modes for the equations of heat flow and wave propagation. However, there are other Hilbert bases for $L^2([0,L])$ that are useful for other purposes such as data compression and image processing. The second video presents a simple form of wavelets, called the Haar basis, that is closely related to how visual images are stored as .jpg files.