Fundamental Solutions for Multivariate Difference Equations

For any function $ a$ with compact support, the $ d$-variate difference equation

$\displaystyle \sum_{k\in\mathbb{Z}^d} a(\ell-k)\,x_k = b_\ell,\quad \forall \ell\in\mathbb{Z}^d\,, $

has a fundamental solution $ x$ (i.e., a solution for $ b = \delta$) with at most power growth, i.e.,

$\displaystyle \vert x_k\vert \le \mathrm{const}\,(1+\Vert k\Vert)^n $

for some $ n\in\mathbb{N}$.

This result is the discrete analogue of Lojasiewicz' theorem (proved also by Hörmander) about the existence of tempered fundamental solutions for constant-coefficient differential operators.


$ \square$ with C. de Boor and S. Riemenschneider, Amer. J. Math. 111 (1989), 403-415.

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