A straight line (y=x) can be written as a sum over sinusoids, known as a Fourier Series.pic.twitter.com/Obii1qd62T
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Replying to @InertialObservr @DynamicsSIAM
No, you only get y=x in (-pi,pi), as you’ve drawn, and then extended periodically from there (since every function in the series is 2pi periodic). The dual of the line is the line, so you’d need an integral to recover the actual line. Presumably you know this, so why fib?...
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Replying to @AlexKontorovich @DynamicsSIAM
that's right, it converges to y=x in (-π,π) for the Fourier series (as opposed to the transform). Given that it does converge to a line on (-π,π), i wouldn't say i fibbed.. This is twitter,not an academic conference.The purpose of my tweets is to get across the basic idea in..
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a neat way and .. if people are curious/want to know more they can pursue it by either asking questions or going to learn it on their own.. Presumably, you know this and that throwing a bunch of conditions at people isn't the best way to engage people
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Replying to @InertialObservr @DynamicsSIAM
I’m all for popularizing fascinating math ideas (wasn’t trying to disparage), of which Fourier series is certainly one, but from reading some of the comments, people seemed rather confused about (and interested in!) what was going on near the endpoints, with good reason...
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there is some detail omitted, as i didn't specify that the boundary conditions enforce the vanishing of the series on the endpoints.. wasn't sure how to make that digestible and wasn't sure if whether or not it'd be helpful trying to explain it in the main tweet
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