What is the rules regarding convergence of a Fourier series?

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SUMMARY

The discussion focuses on the rules for achieving rapid convergence of a Fourier series when only part of a function's period is known. It is established that at least one full period is necessary to fully reconstruct the function. The choice between odd and even extensions is crucial; selecting the extension that results in a continuous function is preferred over one that introduces jump discontinuities. Additionally, the smoothness of the function directly influences the speed of convergence of the Fourier series.

PREREQUISITES
  • Understanding of Fourier series and their convergence properties
  • Knowledge of function continuity and discontinuities
  • Familiarity with odd and even function extensions
  • Basic concepts of L2 convergence in functional analysis
NEXT STEPS
  • Study the properties of Fourier series convergence in detail
  • Learn about odd and even function extensions in Fourier analysis
  • Explore the concept of smoothness and its impact on convergence rates
  • Investigate L2 convergence and its implications in Fourier series
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Mathematicians, engineers, and students studying Fourier analysis, particularly those interested in signal processing and function approximation techniques.

zheng89120
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If you are given part of a period of a Function, what rules would you apply to draw out the full function, so that it converges as quickly as possible as a Fourier series?

thanks
 
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If you only know one part of the period, then there is no way to "draw out the full function." You need at least one full period for that.
 
zheng89120 said:
If you are given part of a period of a Function, what rules would you apply to draw out the full function, so that it converges as quickly as possible as a Fourier series?

thanks

Your question is pretty vague. But if, for example, you have the function defined on (0,p) and you wish to choose between the odd and even extension to (-p,0) to use a half-range expansion, if one of those choices gives a continuous function and the other has a jump discontinuity, the continuous one would be preferred.

Generally, the smoother the function, the quicker the FS converges.
 
What kind of convergence are we assuming L2[a,b], etc?
 

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