I have just started studying fourier series. All the convergences I have seen considered the partial sums to be $\sum\limits_{i=-n}^n a_n Sin(n\theta)$. But in all practical systems the harmonics which have higher magnitude are considered. So my question is if I chose my partial sums such that first I chose harmonic with greatest amplitude(if there is a conflict choose the harmonic with lower frequency) then the next highest and so on.So now will this series of partial sums also converge to same function. If not what all conditions do we need ?
I am guessing that since we are doing infinitely many rearrangements we may need it to be absolutely convergent. But i do not know whether this is really necessary. Any help is appreciated. Thanks
You are right that if the Fourier series is absolutely convergent, then you can do this sort of rearrangement. But in general you should not expect the coefficients of Fourier series to be absolutely convergent. (Nonzero functions for which this does hold are members of what is called the Wiener algebra.) The sum of their squares will be as long as we work with square-integrable functions, but that's a different story.
In fact, even the partial sums of Fourier series will often fail to converge in the usual sense. This is the reason why we deal with alternative notions of summability, like Cesaro summability and Abel summability. The reason Fourier series fail to converge in the usual sense and the ways we get around this are deep ideas in analysis that you can pursue well into graduate school.