I'm having trouble squaring this claim (youtu.be/xWa5_OXI6VM?t=127) with periodic extensions. My understanding is that if we want, for example, the Fourier series for e^x within the interval [0,L], then we can use an even periodic extension, an odd periodic extension, or a "direct" periodic extension (replicating the function's graph from 0 to L), to fill in [−L,0] and make the period 2L.
However, these yield three different Fourier Series (i.e., the even extension will have cosine terms while the odd extension will have sine terms, which won't have a phase shift and thus can't be reconciled). How is this consistent with the claim in the video?
However, these yield three different Fourier Series (i.e., the even extension will have cosine terms while the odd extension will have sine terms, which won't have a phase shift and thus can't be reconciled). How is this consistent with the claim in the video?