  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.
