The Toeplitz theorem and its applications to approximation theory and linear PDEs

We take an algebraic approach to the problem of approximation by dilated shifts of basis functions. Given a finite collection $\Phi$ of compactly supported functions in ${L_p}({\mathbb {R}^s})\quad (1 \leqslant p \leqslant \infty )$, we consider the shift-invariant space $S$ generated by $\Phi$ and the family $({S^h}:h > 0)$, where ${S^h}$ is the $h$-dilate of $S$. We prove that $({S^h}:h > 0)$ provides ${L_p}$-approximation order $r$ only if $S$ contains all the polynomials of total degree less than $r$. In particular, in the case where $\Phi$ consists of a single function $\varphi$ with its moment $\int {\varphi \ne 0}$, we characterize the approximation order of $({S^h}:h > 0)$ by showing that the above condition on polynomial containment is also sufficient. The above results on approximation order are obtained through a careful analysis of the structure of shift-invariant spaces. It is demonstrated that a shiftinvariant space can be described by a certain system of linear partial difference equations with constant coefficients. Such a system then can be reduced to an infinite system of linear equations, whose solvability is characterized by an old theorem of Toeplitz. Thus, the Toeplitz theorem sheds light into approximation theory. It is also used to give a very simple proof for the well-known Ehrenpreis principle about the solvability of a system of linear partial differential equations with constant coefficients.