What is the rules regarding convergence of a Fourier series?

[zheng89120]

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

 

[alexfloo]

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.

 

[LCKurtz]

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.

 

[Bacle]

What kind of convergence are we assuming L²[a,b], etc?