On oscillation of functions with bounded spectral band (Q1873259)

Let Exp be the set of distributions \(f\in S'(\mathbb{R})\) whose Fourier transforms are supported by finite intervals of the real line. Set \[ {\mathcal P}(f):= \{F\in \text{Exp}: \operatorname {supp} \widehat{F} \subset \overline{\text{conv} (\operatorname {supp} \widehat{f})},\;| F|=| f|\text{ on }\mathbb{R}\}. \] The author studies conditions on \(f\in \text{Exp}\) under which there exists \(F\in {\mathcal P}(f) \setminus \{f\}\) with minimal oscillation. The following result is a typical one: Theorem. Let \(f\in \text{Exp}\), \(\overline{\text{supp} (\widehat{f})}= [\sigma,\tau]\), where \(\sigma< \tau\) and \(\sigma\neq -\tau\). If multiplicities of the real zeroes of \(f\) are uniformly bounded, then there exists \(F\in {\mathcal P}(f) \setminus \{f\}\) such that all \(F^{(m)}\) with \(m\geq \sup(\operatorname {mult} f)\) have no real zeroes.