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love the interactive demos. one fundamental thing i always feel is missing with all these videos and articles about the spinny circles set end to end with diff
by enthdegree 7y ago
love the interactive demos.
one fundamental thing i always feel is missing with all these videos and articles about the spinny circles set end to end with different phases and amplitude is: why on earth do such configurations happen to have the capacity to approximate any function you prescribe??
to me this is the entire mystery behind fourier transforms.
the spinny circles are kind of unusual to look at, but do nothing to illuminate to me why convergence of fourier series happens, and for this reason exactly i am of the opinion that this meme analogy is not useful for beginners beyond entertainment.
of course the details for convergence of fourier series are the entire topic of classical harmonic analysis. one hand-wavy way to make sense of it is to first sample and then identify that the dft matrix for the vector space of sampled signals is a basis. kind of similarly, int dx sin x sin nx from 0 to 2pi is 0 for all n, and the span of {sin nx,cos nx} is somehow dense in some function space. although that isn't really very illuminating since to the uninformed it amounts to a numerical coincidence. every single article of this sort that i have seen falls flat in this respect and i feel like this most interesting part has been obscured.
- enthdegree 7y agoit is hard to give a satisfying high level explanation but i would love to see someone try. a Dirichlet kernels proof I just looked up that works for integrable periodic functions is accessible to a really smart high schooler, but still involves a lot of minutae. http://math.uchicago.edu/~may/REU2012/REUPapers/Cuddy.pdf http://math.uchicago.edu/~may/REU2012/REUPapers/Cuddy.pdf . You can define a function called a "n-th order Dirichlet kernel" which when you convolve it with f yields the n-th order Fourier series approximation for f. I guess the missing intuition is that Dirichlet kernels behave more and more like a delta as order goes up, which you can prove using algebra and calculus. Intuitively you know what convolution with a delta looks like, so it is easy to believe that convolution with something near a delta is numerically similar. There is a nice visual on the wiki article for a sequence of Dirichlet kernels looking more and more like a delta.
- salty_biscuits 7y agowhy on earth do such configurations happen to have the capacity to approximate any function you prescribe?? Well they don't. Only certain classes of functions with bounded varitation will have a convergent Fourier series approximation. I think the best way for these demos to introduce this stuff would be to focus on the one with the steps, point out the Gibbs phenomena around the jumps (ringing). Then show for smooth blobby things you are all good though. I think historically a lot of mathematicians had a lot of problems with Fourier's methods applied to the heat equation for these types of reasons (which initial/boundary conditions are ok, etc) hence we have the whole field of harmonic analysis now...