What is the convergence rate of Fourier series?

In summary, the question discussed was about the speed of convergence of Fourier series and what types of functions converge faster than others. It was mentioned that functions with less continuity and similar to a cosine function seem to converge the fastest, but this is not entirely correct. The conversation also touched on the concept of Gibb's phenomenon and its relation to the convergence of Fourier series. It was suggested to look up "convergence rate of Fourier series" for more information.
  • #1
zheng89120
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Okay, so I didn't really understand the professor when he talked about the speed of convergence of Fourier series. The question is what kind of functions converge faster than what kind of other functions using Fourier series representation. My guess from what I have absorbed is that functions with the least continuity and resembles a cosine function the most converges the fastest. Obviously this is not entirely right, and I would like more explanation on this matter.

Thank you.

P.S. How does Gibb's phenomenon tie into the convergence of Fourier series. My guess from what I know is that functions that converge slower will have a higher Gibb's "overshoot".
 
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  • #2

FAQ: What is the convergence rate of Fourier series?

What is the definition of convergence in Fourier series?

The convergence of a Fourier series is the property that describes how closely the partial sums of the series approximate the original function. In other words, as more terms are added to the series, the resulting sum approaches the actual function.

How is the convergence of a Fourier series determined?

The convergence of a Fourier series can be determined using various criteria, such as the pointwise convergence, uniform convergence, or mean square convergence. Each criterion has its own conditions for convergence, and the choice of criterion depends on the properties of the function being approximated.

What is the significance of convergence in Fourier series?

The convergence of a Fourier series is important because it guarantees that the partial sums of the series will approach the original function as the number of terms increases. This allows for accurate approximations of functions that are difficult to represent directly.

Can a Fourier series converge to a function that is not the original function?

Yes, it is possible for a Fourier series to converge to a different function than the original function being approximated. This is known as Gibbs phenomenon, where the partial sums of the series have large oscillations near the discontinuity points of the function.

Are there any known functions for which the Fourier series does not converge?

Yes, there are certain functions for which the Fourier series does not converge. These include functions with discontinuities, infinite discontinuities, or unbounded derivatives. In these cases, alternative methods of approximation, such as the use of generalized Fourier series, may be necessary.

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