A Fourier series refresher.
Posted by peeterjoot on May 3, 2012
I’d used the wrong scaling in a Fourier series over a interval. Here’s a reminder to self what the right way to do this is.
Suppose we have a function that is defined in terms of a trigonometric Fourier sum
where the domain of interest is . Stating the problem this way avoids any issue of existence. We know exists, but just want to find what they are given some other representation of the function.
Multiplying and integrating over our domain we have
We want all the terms in the sum to be be zero, requiring equality of the exponentials, or
This fixes our Fourier coefficients
Given this, the correct (but unnormalized) Fourier basis for a interval would be the functions , or the sine and cosine equivalents.