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