Abstract
Let E(x, ξ, t) be the Mehler kernel and let U(x, t) be the solution, with some appropriate growth estimate (*), of the Hermite heat equation in R×(0, T). Then there exists a unique Fourier hyperfunction u such that U(x, t)=⟨ u(ξ), E(x, ξ, t)⟩. Conversely for any Fourier hyperfunction u, U(x, t):=⟨ u(ξ), E(x, ξ, t)⟩ is the smooth solution of the Hermite heat equation in R×(0, T) satisfying the growth estimate (*).