Oscillation of Fourier transform and Markov-Bernstein inequalities

Under certain conditions on an integrable function f having a real-valued Fourier transform Tf=F, we obtain a certain estimate for the oscillation of F in the interval [-C||f'||/||f||,C||f'||/||f||] with C>0 an absolute constant. Given q>0 and an integrable positive definite function f, satisfying some natural conditions, the above estimate allows us to construct a finite linear combination P of translates f(x+kq)(with k running the integers) such that ||P'||>c||P||/q, where c>0 is another absolute constant. In particular, our construction proves sharpness of an inequality of H. N. Mhaskar for Gaussian networks.