A New Constructive Proof of the Malgrange-Ehrenpreis Theorem

see [1, Thm. 6, p. 892], [6, Thm. 1, p. 288]. This theorem was a first impressive piece of evidence of the impact of distribution theory in its application to linear partial differential equations, and several different proofs of it were therefore devised in the subsequent years, see [3, Ch. X, and the historic notes in pp. 58, 59]. An elementary proof based on L2-theory was presented in [9]. For more details, we refer to the surveys in [7], [8], [13]. Already very early, constructive proofs were also found, i.e., proofs that represent a fundamental solution E by a formula, instead of inferring the existence of E from the Hahn–Banach theorem as did the original proofs. The prototype of such a formula is “Hörmander’s staircase,” which employs partitions of unity, [11], [7, (1), p. 345], [8, Thm. 2.3, p. 107]. While these partitions of unity are based on the location of the zeroes of P(z), z ∈ Cn, and the formula is not completely explicit in this sense, H. König published in 1994 a proof using a representation of a fundamental solution by an n-fold integral with respect to η of the inverse Fourier transforms F−1( fη) of the modulus-one functions fη(ξ) = P(iξ + η)/P(i ξ + η), see [5], [8, Thm. 2.5, p. 111]. A simpler formula involving a one-fold integral was given in [7, (2), p. 346]. In this paper, we shall further simplify this procedure by constructing a fundamental solution as a sum over finitely many distributions F−1( fη j ), j = 0, . . . ,m = deg P, see Proposition 1. In a second part, we shall then generalize this formula so as to obtain a constructive proof of the existence of a fundamental solution for linear partial differential-difference equations, i.e., convolution equations with kernels of finite support, see Proposition 2. Let us introduce next some notation. The letter n is reserved for the dimension of the underlying vector space Rn. The inner product on Rn is indicated by juxtaposition, i.e., ξ x = ξ1x1 + · · · + ξn xn. In this paper, all partial differential operators are linear and contain constant complex coefficients, i.e., they belong to C[∂] = C[∂1, . . . , ∂n], and we adopt for them the usual multi-index notation. In particular, we