The Toeplitz Theorem and its Applications to Approximation Theory and Linear PDE’s

We take an algebraic approach to the problem of approximation by dilated shifts of basis functions. Given a finite collection Φ of compactly supported functions in Lp(IR) (1 ≤ p ≤ ∞), we consider the shift-invariant space S generated by Φ and the family {S : h > 0}, where S is the h-dilate of S. We prove that {S} provides Lp-approximation order r only if S contains all the polynomials of total degree less than r. In particular, in the case where Φ consists of a single function φ with its moment ∫ φ 6= 0, we characterize the approximation order of {S} 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 the structure of 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. The Toeplitz theorem gives a necessary and sufficient condition for an infinite system of linear equations to have solutions. Thus, the Toeplitz theorem sheds new insight into approximation theory. It is also used to give a very simple proof for the wellknown Ehrenpreis principle about the solvability of a system of linear partial differential equations with constant coefficients.