On the division problem for a tempered distribution that depends holomorphically on a parameter

Abstract

We give sufficient conditions ensuring a construction of solution to the equation

$$\sum\limits_{k = 0}^m {{P_k}\left( \sigma \right){\lambda ^k}u\left( \lambda \right) = f\left( \lambda \right)} $$

with σ ∈ ℝⁿ and λ ∈ G ⊂ ℂ, where f(λ) and u(λ) are tempered distributions depending holomorphically on λ, while the polynomial P _m (σ) may have real zeros.