Let u be a real valued strong solution defined in a cylindrical domain of a linear second-order parabolic equation in two space variables with entire coefficients. Then it is shown that on compact subsets of its domain of definition u can be approximated arbitrarily closely in the maximum norm by an entire solution of the parabolic equation.

Though it is still an open problem for which class of first-order elliptic systems Carleman's theorem holds, this is proven here for a certain class of systems (with analytic coefficients) for which Douglis introduced the hypercomplex algebra and hyperanalytic functions. The proof is based on a representation formula generalising Vekua's approach with Volterra integral equations in C2 to more than two unknowns. The representation formula is of its own interest because it provides the generation of complete families of solutions. The equations of plane inhomogeneous elasticity problems lead to a system of the desired class.