Preservation of Classes of Entire Functions under Pure Imaginary Perturbations of Their Zero Sets

The problem studied in this paper concerns with perturbations of zero sets for entire functions. Namely, we consider weighted spaces of entire functions \(\mathcal{P}\) that are Fourier–Laplace transform images of spaces of \(\Omega\) -ultradifferentiable test functions or \(\Omega\) -ultradistributions and their subsets \(M(\mathcal{P}),\) \(\mathcal{D}(\mathcal{P})\) formed by multipliers or divisors of \(\mathcal{P}\) , respectively. Our main purpose is to prove the assertion: given a function \(f\) in \(\mathcal{P}\) , \(M(\mathcal{P})\) or \(\mathcal{D}(\mathcal{P})\) , pure imaginary perturbations of its zeros preserve \(f\) in the corresponding function class.