Division - Interpolation methods and Nullstellensätze Carlos

The Fundamental Principle of Ehrenpreis asserts that any solution (in an appropiate functional space, which we will assume, to simplify, is the space of C∞ functions in R) of a system of homogeneous linear partial differential equations with constant coefficients in R, n ≥ 2, can be can be represented in terms of the exponential polynomials solutions of the system [Ehr, Pal, Bjo]. Another way to phrase the Fundamental Principle is to say that in certain spaces of entire functions with restricted growth in C one has an explicit linear (and continuous) division algorithm with remainder with respect to the ideal generated by the symbols of these equations. Such an assertion, even the fact the exponential polynomial solutions of the system are dense in the space of all solutions, fails for systems of homogeneous convolution equations μ1 ∗ f = · · · = μm ∗ f = 0 [Gu]. Moreover, there are few positive examples of transcendental nature where we know that the Fundamental Principle (or just Spectral Synthesis) holds. Note that it is still possible to state this principle by saying that in the Paley-Wiener algebra there is an explicit linear and continuous division algorithm with remainder with respect to the ideal generated by the Fourier transforms μ̂j of the convolution operators. In particular, the ideal generated by these Fourier transforms should be closed. By extension, we will say that a system of convolution equations