Under certain conditions on an integrable function P having a real-valued Fourier transform P@^ and such that P(0)=0, we obtain an estimate which describes the oscillation of P@^ in [-C@?P^'@?"~/@?P@?"~,C@?P^'@?"~/@?P@?"~], where C is an absolute constant, independent of P. Given @l>0 and an integrable function @f with a non-negative Fourier transform, this estimate allows us to construct a finite linear combination P"@l of the translates @f(.+k@l),k@?Z, such that @?P"@l^'@?"~>c@?P"@l@?"~/@l with another absolute constant c>0. In particular, our construction proves the sharpness of an inequality of Mhaskar for Gaussian networks.
['Szilárd Gy. Révész', 'Noli N. Reyes', 'Gino Angelo M. Velasco']
Oscillation of Fourier transforms and Markov--Bernstein inequalities