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

In summary, the conversation is about calculating the error between a function and its Fourier series representation based on the partial sums of the series and exploring how well a Fourier series can approximate angles. The error is expected to approach a non-zero value as the number of terms increases, and the Gibbs phenomenon is relevant in cases of discontinuous functions.
  • #1
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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  • #2
Approximate angles? The Gibbs phenomenom occurs where the function is discontinuous.
 

What is the concept of "Error in Fourier series"?

The "Error in Fourier series" refers to the difference between a given function and its Fourier series approximation. It represents the discrepancy between the exact value of the function and the value calculated using a finite number of terms in the Fourier series.

What is the significance of studying the "Error in Fourier series"?

Studying the "Error in Fourier series" allows us to understand the limitations of approximating a function using a finite number of terms in the Fourier series. It also helps us to determine the accuracy of the approximation and improve it by adding more terms.

How is the "Error in Fourier series" calculated?

The "Error in Fourier series" can be calculated using various methods such as the root mean square error, the maximum error, or the mean absolute error. These methods involve comparing the values of the function and its Fourier series approximation at different points and calculating the difference between them.

What factors affect the "Error in Fourier series"?

The "Error in Fourier series" is affected by the smoothness of the function, the number of terms used in the Fourier series, and the interval over which the function is approximated. A function with sharp corners or discontinuities will have a larger error compared to a smooth function. Similarly, using a larger number of terms in the Fourier series can reduce the error, but it also increases the computational complexity.

How can the "Error in Fourier series" be minimized?

The "Error in Fourier series" can be minimized by using more terms in the Fourier series, choosing a smaller interval for approximation, or using a different series representation such as a trigonometric or exponential series. Additionally, using numerical methods such as least squares approximation can also help to reduce the error.

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