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does anyone know if there is a book that deal with series of the kind, $$\displaystyle \sum_{\xi\in\mathbb Z^n}a_\xi,$$ that is, when the indices are in the space $\mathbb Z^n$. I'm looking for the theory of convergence of these series.. Thanks

PtF
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    I'm a little confused. First and foremost, to talk of convergence, you need to fix an ordering of $\mathbb{Z}^n$. If this sequence (with the fixed ordering) has absolutely convergent sum, then it converges in any order and your notation makes sense. But then, with this definition of your sum, this is nothing but a standard infinite series. – Alex Youcis Aug 22 '13 at 17:52
  • you mean I can apply the standard tests for checking the convergence of such a series? – PtF Aug 22 '13 at 18:26
  • May you help us about the background, where does this come from, what is the context. That would help to have a more concrete outline to support you. – al-Hwarizmi Aug 22 '13 at 18:34
  • Well @al-Hwarizmi this comes from periodic Fourier analysis.. If $f\in C^\infty(\mathbb T^n)$, where $\mathbb T^n$ is the $n$-torus then we can write its Fourier series representation $$f(x)=\sum_{\xi\in\mathbb Z^n}\hat{f}(\xi)e^{2\pi ix\cdot \xi}.$$ I came across this kind of things in my master thesis, which I'm still developing, about pseudo-differential operators on torus...In the proof of many theorem I must know about the convergence of series indexed on $\mathbb Z^n$... – PtF Aug 22 '13 at 21:02

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First of all, there is no canonical ordering of $\mathbb{Z}^n$. But $\mathbb{Z}^n \cong \mathbb{Z}$ as sets. Only the cardinality of the index set plus the ordering is decisive.

There are places, where you want to consider $\mathbb{Z}^n$ as a subset of $\mathbb{R}^n$ though, as I can see from your comment. This is important e.g. for applying the Poisson summation formula. With it, you can for example compute the number of points $(k_1, \dots, k_n)$ with $k_1^2 + \dots + k_n^2 \leq D$ (Gauss circle problem) or analytically continue L-functions.

Marc Palm
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