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...

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Bibliographic Details
Published in:arXiv.org 2006-03
Main Authors: Revesz, Szilard Gy, Reyes, Noli N, Velasco, Gino Angelo M
Format: Article
Language:English
Subjects:
Fourier transforms
Integers
Markov processes
Mathematical functions
Sharpness
Online Access:Get full text
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Summary: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.
ISSN:2331-8422
DOI:10.48550/arxiv.0603346