# Terminology for multidimensional Fourier series

Dear All, I'm computing multidimensional Fourier series of a function $f$ defined on $(0, L_1)\times(0, L_2)\times\cdots\times(0,L_d)$. The series reads

$f(\vec x)=\sum_{\vec k}\hat f(\vec k)\exp(\imath\vec k\cdot\vec x)$

where the sum is extended to all

$\vec k = \dfrac{2\pi a_1}{L_1}\vec e_1+\cdots+\dfrac{2\pi a_d}{L_d}\vec e_d\qquad(a_1, \ldots, a_d\in\mathbb Z).$

My question is: how do you call the set of the vectors $\vec k$. I think specialists in diffraction would call this set "reciprocal lattice"; how about mathematicians?

Thanks a lot in advance, Sebastien

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It's usually called the dual lattice $\hat{\Lambda}$ of the lattice $\Lambda = L_1 \mathbb{Z} \oplus L_2 \mathbb{Z} \oplus \cdots \oplus L_{d} \mathbb{Z}$ in $\mathbb{R}^d$. See this Wikipedia page on the reciprocal lattice for further explanation and have also a look at the one on Pontryagin duality.