How to Calculate Error in Fourier Series and its Approximation of Angles?

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SUMMARY

This discussion focuses on calculating the error between a function and its Fourier series representation, particularly in relation to the series' partial sums. The Gibbs phenomenon is highlighted as a critical concept, especially in contexts where the function exhibits discontinuities. The user expresses interest in understanding how the error behaves as the number of terms in the Fourier series increases, particularly concerning angle approximations. The conclusion drawn is that the error does not diminish to zero for discontinuous functions, which is a key insight for further exploration.

PREREQUISITES
  • Fourier series theory
  • Understanding of the Gibbs phenomenon
  • Concept of partial sums in series
  • Basic calculus and continuity of functions
NEXT STEPS
  • Research the Gibbs phenomenon in detail
  • Explore methods for calculating error in Fourier series approximations
  • Study the convergence properties of Fourier series
  • Investigate applications of Fourier series in approximating angular functions
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Mathematicians, engineers, and students studying signal processing or harmonic analysis who are interested in Fourier series and their error characteristics.

Sandalwood
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Does anyone know how to calculate the error between a function and its Fourier series representation as a function of the partial sums of the series? So far I haven't been able to find anything in the literature that talks about this.

I'm also interested in looking at how well a Fourier series can approximate angles. My guess is that the error approaches a non-zero value at the angle as you take an infinite number of terms because every term in a Fourier series is smooth and continuous. I just stumbled across the Gibbs phenomenon after a Google search, so I'll be looking into that. But if anyone has anything else to add, I'd appreciate it.

Thanks in advance.
 
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Approximate angles? The Gibbs phenomenom occurs where the function is discontinuous.
 

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