Physics 434, 2012: Lecture 17

From Ilya Nemenman: Theoretical Biophysics @ Emory
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Back to Physics 434, 2012: Information Processing in Biology. Here we introduce the idea of Fourier series and Fourier transforms. We have discussed in the previous lecture why we need them.

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  1. Consider a function periodic on .
  2. We would like to approximate this function as , takin at some point.
  3. From this expression, we can find the coefficients self-consistently. Indeed, let's multiply the equation by and integrate from to .
  4. All terms containing products are zero.
  5. For the terms, we have .
  6. Completing the integrals, we have: .
  7. We can do similar to find : multiply by , and integrate.
  8. This gives: , , .