Poisson, Guinand, Meyer

[Posted on march 27, 2016]

A recent highly interesting paper by Yves Meyer (PNAS paywalled, local version at ENS Cachan, and seminar notes) constructs explicitly new Poisson-type summation formulas (building on previous little known work of Andrew-Paul Guinand  and an existence result of Nir Lev and Aleksander Olevskii) : the big difference with Poisson summation is that the new formulas do not have support on a lattice but only on a locally finite set (and then provide new examples of crystalline measures).

Since these new results involve some arithmetic (see below) I’ve asked over at MO whether this was known to number theorists, but there hasn’t been any immediate answer, so perhaps not and there’s probably room for interesting further work on the topic.

To state things very explicitely (for my own benefit, but also just for the beauty of it), here are the formulas taken directly from Meyer’s paper :

Poisson (Dirac comb case): on a lattice \Gamma\subset\mathbb{R}^n and its dual \Gamma^* we have for any function f in the Schwartz class \mathcal{S}(\mathbb{R}^n) that

\displaystyle \mbox{vol}(\Gamma) \sum_{\gamma\in\Gamma}f(\gamma) = \sum_{\eta \in\Gamma^*}\widehat{f}(\eta)

Poisson (corollary of Dirac comb case) : for every \alpha,\beta \in\mathbb{R}^n we have (in terms of distributions to make the comb more explicit still)

\displaystyle \mbox{vol}(\Gamma) \sum_{\gamma\in\Gamma +\alpha} e^{2i\pi\beta .\gamma}\delta_{\gamma} = e^{2i\pi\alpha .\beta} \sum_{\eta \in\Gamma^* +\beta} e^{2i\pi\alpha .\eta}\delta_{\eta}

Guinand : define for any n\in\mathbb{N} the number of sums of three squares that equal to n by r_3(n) (by Legendre’s theorem this is possible only for those n not of the form 4^j(8k+7)). Then introducing Guinand’s distribution (acting on functions of the variable t)

\displaystyle \sigma := -2\frac{d}{dt}\delta_0 + \sum_{n=1}^{+\infty} \frac{r_3(n)}{\sqrt{n}} (\delta_{\sqrt{n}}-\delta_{-\sqrt{n}})

then we have \langle \sigma ,f\rangle = \langle -i\sigma ,\widehat{f}\rangle .

Meyer (first example) : introducing the function \chi on \mathbb{N} (this is a clash of notation with Dirichlet characters) by \chi(n)=-\frac{1}{2} when n\not\equiv 0\pmod{4}, \chi (n)=4 when n\equiv 4\pmod{16} and \chi (n)= 0 when n\equiv 0\pmod{16} then with the distribution

\displaystyle \tau := \sum_{n=1}^{+\infty} \frac{\chi(n)r_3(n)}{\sqrt{n}} (\delta_{\frac{\sqrt{n}}{2}}-\delta_{-\frac{\sqrt{n}}{2}})

we have \langle \tau ,f\rangle = \langle -i\tau ,\widehat{f}\rangle .

The support of \sigma and \tau are thus defined as subsets of \{\pm\sqrt{n}|n\in\mathbb{N}\} and \{\pm\frac{\sqrt{n}}{2}|n\in\mathbb{N}\} by their respective arithmetic conditions, and thus are definitely not equally spaced lattice points.

Meyer (second example) : with the distribution

\displaystyle \rho := 2\pi\delta_{\frac{1}{2}} +2\pi\delta_{-\frac{1}{2}} + \sum_{n=1}^{+\infty} \frac{\sin(\pi\sqrt{n})r_3(n)}{\sqrt{n}} (\delta_{\frac{\sqrt{n}}{2}+\frac{1}{2}}+\delta_{\frac{\sqrt{n}}{2}-\frac{1}{2}}+\delta_{-\frac{\sqrt{n}}{2}+\frac{1}{2}}+\delta_{-\frac{\sqrt{n}}{2}-\frac{1}{2}} )

we have \langle \rho ,f\rangle = \langle \rho ,\widehat{f}\rangle (very nice!).

There are several other examples in Meyer’s paper, as well as higher-dimensional constructions (that I haven’t absorbed yet, so I’ll stop here).

Update (march 27): two relevant papers I’ve just found

  • On the Number of Primitive Representations of Integers as Sums of Squares by Shaun Cooper and Michael Hirschhorn published in Ramanujan J (2007) 13:7–25, which in particular provides the explicit formula r_3(n)=\sum_{d^2|n}r_3^p\big ( \frac{n}{d^2}\big ) where the function r_3^p is in turn explicited (p is a label standing for ‘primitive’)
  • Irregular Poisson Type Summation by Yu. Lyabarskii and W.R. Madych, published in SAMPLING THEORY IN SIGNAL AND IMAGE PROCESSING Vol. 7, No. 2, May 2008, pp. 173-186, which does prove a Poisson-type formula with irregularly spaced sampling points (but if I understand well the examples they mention at the end show it is still different from the results of Guinand and Meyer, to be confirmed)
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