Oscillation of Fourier transforms and Markov–Bernstein inequalities

Abstract

Under certain conditions on an integrable function P having a real-valued Fourier transformP and such that P (0)=0, we obtain an estimate which describes the oscillation ofP in [−C P ∞ / P ∞ , C P ∞ / P ∞ ], where C is an absolute constant, independent of P. Given > 0 and an integrable function with a non-negative Fourier transform, this estimate allows us to construct a finite linear combination P of the translates (· + k ), k ∈ Z, such that P ∞ > c P ∞ / with another absolute constant c > 0. In particular, our construction proves the sharpness of an inequality of Mhaskar for Gaussian networks.