Oscillation of Fourier transform and Markov-Bernstein inequalities
Classical Analysis and ODEs
2007-05-23 v1
Abstract
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.
Cite
@article{arxiv.math/0603346,
title = {Oscillation of Fourier transform and Markov-Bernstein inequalities},
author = {Szilard Gy. Revesz and Noli N. Reyes and Gino Angelo M. Velasco},
journal= {arXiv preprint arXiv:math/0603346},
year = {2007}
}