On the stability of interval families of characteristic functions depending linearly on parameters (Q1342357)

Let \(D\) be a closed region in \(\mathbb{C}\) with a piecewise smooth boundary and \(P= \Pi P_ j\) a closed region in a real finite dimensional topological Hausdorff space. Let \(F\) be the parametric family of holomorphic functions \(\{f_ p (z)\}\) in \(D\), \(p\in P\), such that \(\forall f_ i(z)\in F\) is continuous in \(p\) and \[ f_ i(z)= f_ 0(z)+ p_ 1 f_ i(z)+ \dots+ p_ n f_ n(z). \] The author has obtained the conditions which guarantee that \(f_ p(z) \neq0\), \(z\in D\), \(p\in P\). Besides the generalization of Kharitonov's theorem for exponential polynomials is proved.