Local approximation of arbitrary functions by solutions of nonlocal equations

TL;DR

This paper demonstrates that any function can be locally approximated by solutions to nonlocal linear equations, including non-elliptic and non-parabolic types, with applications to s-caloric functions.

Contribution

It introduces a method for local approximation of arbitrary functions using solutions of nonlocal equations, extending to non-elliptic and non-parabolic operators.

Findings

Any function can be approximated locally by solutions of nonlocal equations.

Every function is locally s-caloric up to a small error.

The approach applies to non-elliptic and non-parabolic operators.

Abstract

We show that any function can be locally approximated by solutions of prescribed linear equations of nonlocal type. In particular, we show that every function is locally $s$-caloric, up to a small error. The case of non-elliptic and non-parabolic operators is taken into account as well.