Treffer: Infinite Order Differential Operators in Spaces of Entire Functions

Differential operators ϕ(∆θ,ω), where ϕ is an exponential type entire function of a single complex variable and ∆θ,ω = (θ+ωz)D+zD 2, D = ∂/∂z, z ∈ C, θ ≥ 0, ω ∈ R, acting in the spaces of exponential type entire function are studied. It is shown that, for ω ≥ 0, such operators preserve the set of Laguerre entire functions provided the function ϕ also belongs to this set. The latter consists of the polynomials possessing real nonpositive zeros only and of their uniform limits on compact subsets of the complex plane C. The operator exp(a∆θ,ω), a ≥ 0 is studied in more details. In particular, it is shown that it preserves the set of Laguerre entire functions for all ω ∈ R. An integral representation of exp(a∆θ,ω), a> 0 is obtained. These results are used to obtain the solutions to certain Cauchy problems employing ∆θ,ω.