Malgrange Theorems for Infinite-Dimensional Differential Operators

Abstract

We prove Malgrange-type existence and approximation theorems for partial differential operators and spaces in the ring of formal power series ≡ N(N) C, N (N) ≡⊕ ∞ N, in an infinite number of variables. In particular we study spaces of entire functions within this framework. With the infinite-dimensional Fourier–Borel transform as a tool, we prove existence theorems for the spaces , ,A (X), Exp(Y ) and F. Here ≡⊕ N(N) C is the space of finitely supported polynomials, A(X) and Exp(Y ) are spaces of entire (respectively exponential-type) functions and F is the Fischer–Fock (Hilbert) space. These spaces are related as follows: ⊆ Exp(Y ) ⊆ F ⊆ A(X) ⊆ , and can, pairwise, be considered as dual to one another. The key result for the existence theorem on F is a division theorem for the spaces Exp(Y ) ,A (X) and F. Furthermore, we show that homogeneous solutions can be approximated by homogeneous solutions consisting of exponential finitely supported polynomials.