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Determine the real Fourier series for $f(t) = t^2$ with $t \in [0, 1[$.

How would you periodically expand this so that the Fourier series converges as fast as possible? Would this be a good expansion?

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Savantik
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  • What's the period of the function $f$? The coefficients heavily depend on how $f$ is defined and where. If this question was given to you as is, I would assume that the function of interest is specifically the $1$-periodic extension of $f$. – Bruno B Jun 07 '23 at 18:27
  • The period of the function f is T = 1, with my expansion the period is 2 now. I just don't really know how to pick the right periodic expansion for the fastest converging series. How do I know what to expand to? – Savantik Jun 07 '23 at 18:30
  • If someone wants you to calculate the Fourier series of a given function, you can't just decide to change functions... But if the question in the title is not your true question, and you really want an answer to your second question, you should probably change the title and make it the real focus of the post. – Bruno B Jun 07 '23 at 18:38
  • Yes, I now changed the question to what if the Fourier series has to converge as fast as possible, what would we do with this case? – Savantik Jun 07 '23 at 18:43
  • @BrunoB I believe what Savantik refers to is whether to extend the function periodically from the half period 1 to the full period of 2 as an odd function (and thus a sine series) or as an even function, which is being pictured (and thus a cosine series) Vs just expanding as a cosine and sine series for half period. Although I personally know nothing about the speed of convergence of these different approaches. – Theo Diamantakis Jun 07 '23 at 19:00
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    Yes, the question needs to be phrased much more carefully. Are you comparing rate of convergence of the $1$-periodic Fourier expansion with that of the various possible $2$-periodic extensions? $3$-periodic? … – Ted Shifrin Jun 07 '23 at 19:02
  • Might be relevant https://math.stackexchange.com/a/10854/27978 – copper.hat Jun 07 '23 at 20:06

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