Characterization of Fourier hyperfunctions by solutions of the Hermite heat equation

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 (*).

Keywords:

• Hermite functions
• Mehler kernel
• Hermite heat equation