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In Fourier analysis, how should I interpret sums of the form $$ \sum_{n \in \mathbb{Z}} a_n = \sum_{n=-\infty}^\infty a_n?$$ Is it $$\lim_{N \to \infty} \sum_{|n| < N} a_n$$ or $$\sum_{n=1}^\infty a_{-n} + \sum_{n=0}^\infty a_n?$$ When do I not have to care about the order of summation?

user369210
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  • I would use the latter interpretation. – copper.hat Dec 27 '16 at 19:13
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    I have most often seen the former used in discussion above convergence of Fourier series (for example as seen here https://en.wikipedia.org/wiki/Convergence_of_Fourier_series). – Winther Dec 27 '16 at 19:38
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    Note that summation order is irrelevant if and only if either $\sum_{n \in \mathbb{Z}} \max[a_n,0] < \infty$ or $\sum_{n \in \mathbb{Z}} \min[a_n,0] > -\infty$. (This allows the sum to be $\infty$ or $-\infty$). If you want to ensure a finite sum that is independent of order, then you need $\sum_{n\in \mathbb{Z}} |a_n| < \infty$. – Michael Dec 27 '16 at 20:36

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The first is sometimes called the principal value since it is similar to the Cauchy Principal Value for sums. The second is the analog of the standard convergence of integrals. In any case, describing the kind of convergence is best.

The sum $$ \pi\cot(\pi z)=\sum_{k\in\mathbb{Z}}\frac1{k+z} $$ requires the principal value sum to converge, but it is often written as above.

robjohn
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