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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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?...
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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Is teaching kids Newtonian mechanics fibbing, or should we also tell them that it's only valid in the domain of weak gravitational force (i mean spacetime curvature).. but actually /that's/ not quite true when we consider QM as well.. i wish you good luck with that curriculum
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But ... isn't the "basic idea" that one can use a Fourier series to approximate any periodic signal? So why start with a "straight line", an inherently aperiodic object? Also, sorry for the nitpick, but it doesn't converge to y=x in (-π,π). The integral error converges to zero.
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lines are precisely what sawtooth waves are..
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