Oscillation of Fourier transforms and Markov-Bernstein inequalities

JOURNAL OF APPROXIMATION THEORY

卷:145

Article

Revesz, Szilard Gy. ; Reyes, Noli N. ; Velasco, Gino Angelo M.

关键词: oscillation of Fourier transform;    Markov-Bernstein inequalities;    sums of translates;    Gaussian networks;   

DOI  :  10.1016/j.jat.2006.07.004

来源: Elsevier

【 摘 要 】

Under certain conditions on an integrable function P having a real-valued Fourier transform (P) over cap and such that P(0)=0, we obtain an estimate which describes the oscillation of (P) over cap in [-C parallel to P'parallel to(infinity)/parallel to P parallel to(infinity), C parallel to P'parallel to(infinity)/parallel to P parallel to(infinity)], where C is an absolute constant, independent of P. Given lambda > 0 and an integrable function phi with a non-negative Fourier transform, this estimate allows us to construct a finite linear combination P-lambda of the translates phi(. + k lambda), k is an element of Z, such that parallel to P-lambda'parallel to infinity > c parallel to P-lambda parallel to(infinity)/lambda with another absolute constant c > 0. In particular, our construction proves the sharpness of an inequality of Mhaskar for Gaussian networks. (C) 2006 Elsevier Inc. All rights reserved.

【 授权许可】

Free