Trigonometric Convex Underestimator for the Base Functions in Fourier Space

A three-parameter (a, b, x_s) convex underestimator of the functional form (x) = -a sin[k(x-x_s)] + b for the function f(x) = sin(x+s), x [x_L, x_U], is presented. The proposed method is deterministic and guarantees the existence of at least one convex underestimator of this functional form. We show that, at small k, the method approaches an asymptotic solution. We show that the maximum separation distance of the underestimator from the minimum of the function grows linearly with the domain size. The method can be applied to trigonometric polynomial functions of arbitrary dimensionality and arbitrary degree. We illustrate the features of the new trigonometric underestimator with numerical examples.Support from the National Science Foundation and the National Institutes of Health Grant R01 GM52032 is gratefully acknowledged.