Oscillation of Fourier transforms and Markov-Bernstein inequalities

Under certain conditions on an integrable function P having a real-valued Fourier transform P̂ and such that P(0)=0, we obtain an estimate which describes the oscillation of P̂ in [−C‖P ′‖∞/‖P ‖∞, C‖P ′‖∞/‖P ‖∞], whereC 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. © 2006 Elsevier Inc. All rights reserved.