What is the rules regarding convergence of a Fourier series?

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Discussion Overview

The discussion revolves around the rules and considerations for achieving convergence of a Fourier series when only part of a function's period is known. It explores the implications of function extensions and the nature of convergence in this context.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant suggests that to draw out the full function from part of a period, certain rules should be applied to ensure quick convergence of the Fourier series.
  • Another participant argues that without knowledge of at least one full period, it is impossible to accurately reconstruct the full function.
  • A further reply indicates that if a function is defined on a specific interval and one is choosing between odd and even extensions for a half-range expansion, the choice that results in a continuous function is preferable over one that leads to a jump discontinuity.
  • Another participant questions what type of convergence is being assumed, mentioning L2[a,b] as a potential consideration.

Areas of Agreement / Disagreement

Participants express differing views on the feasibility of reconstructing a function from partial information, and there is no consensus on the specific rules or types of convergence applicable in this scenario.

Contextual Notes

The discussion highlights limitations related to the assumptions required for function reconstruction and the definitions of convergence being referenced, which remain unresolved.

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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