Density of polynomial elements in invariant subspaces of entire functions of exponential type
When studying the approximation problem for arbitrary solutions (in different classes of functions) of a differential equation with constant coefficients by polynomial solutions, Malgrange found a necessary and sufficient condition for this approximation, namely, the irreducible components of the characteristic polynomial of the differential equation must vanish at the origin. Later on, Palamodov proved for systems of differential equations that every solution can be approximated by exponential polynomials of the form \(p(z)\exp(z,\zeta)\), \(\zeta\in B\), where the set \(B\supset\mathbb C^n\) has nonempty intersection with every manifold in the corresponding family of Noetherian operators and manifolds. The objective of the present paper is the extension of the Malgrange and Palamodov results to translation invariant subspaces in the space of entire functions of exponential type of several complex variables. Conditions are presented under which the set of exponential polynomials is dense in such a space. In particular, necessary and sufficient conditions for the polynomial elements to be dense in invariant subspaces are obtained.
scientific article; zbMATH DE number 2121367