Jeffrey D. Vaaler

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We determine extremal entire functions for the problem of majorizing, minorizing, and approximating the Gaussian function e−πλx 2 by entire functions of exponential type. The combination of the Gaussian and a general distribution approach provides the solution of the extremal problem for a wide class of even functions that includes most of the previously(More)
Given a monic real polynomial with all its roots on the unit circle, we ask to what extent one can perturb its middle coefficient and still have a polynomial with all its roots on the unit circle. We show that the set of possible perturbations forms a closed interval of length at most 4, with 4 achieved only for polynomials of the form x2n + cxn + 1 with c(More)
We obtain extremal majorants and minorants of exponential type for a class of even functions on R which includes log |x| and |x|α, where −1 < α < 1. We also give periodic versions of these results in which the majorants and minorants are trigonometric polynomials of bounded degree. As applications we obtain optimal estimates for certain Hermitian forms,(More)