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arXiv · 2412.19539

On the approximation of spatial convolutions by PDE systems

Hiroshi Ishii·Yoshitaro Tanaka
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Abstract

This paper develops an approximation framework for spatial convolutions with radial kernels using elliptic PDE systems in arbitrary spatial dimensions. We prove that radial kernels in Sobolev spaces can be approximated arbitrarily well by linear sums of Green functions associated with a prescribed sequence of positive parameters. The proof is based on a completeness result for an auxiliary system of linear sums of Green functions, established through the identity theorem for analytic functions and a uniqueness argument by integral transforms. The Sobolev approximation provides control of derivatives of the kernels and yields corresponding approximations of convolution operators. As an application, we show that solutions of an aggregation-confinement-diffusion equation can be approximated on any finite time interval by solutions of a corresponding local parabolic-elliptic Keller-Segel system. A numerical example illustrates the kernel approximation.

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Hiroshi Ishii, Yoshitaro Tanaka. 2026-09-19. On the approximation of spatial convolutions by PDE systems. https://arxiv.org/abs/2412.19539

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