Weight index of concentration

In terms of the concentration weight index, the behavior of the function |W(reiθ)|-1 is investigated as θ→0, where W is an even entire function of exponential type that has only real zeros. This question is relevant in a number of problems of complex analysis related to the strongly nonspanning (strongly minimality) of a system of exponentials on a family of curves, Pavlov-Korevar-Dixon interpolation, and analytic continuation of limit functions of sequences of polynomials from exponentials. This circle of problems goes back to the following problem of A. F. Leontief posed in 1956: under what conditions supθ≠0,πH(θ)