Oscillation of Fourier transforms and Markov–Bernstein inequalities

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 λ...

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Bibliographic Details
Published in:Journal of approximation theory 2007-03, Vol.145 (1), p.100-110
Main Authors: Révész, Szilárd Gy, Reyes, Noli N., Velasco, Gino Angelo M.
Format: Article
Language:English
Subjects:
Gaussian networks
Markov–Bernstein inequalities
Oscillation of Fourier transform
Sums of translates
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Summary: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 λ > 0 and an integrable function φ with a non-negative Fourier transform, this estimate allows us to construct a finite linear combination P λ of the translates φ ( · + k λ ) , k ∈ Z , such that ∥ P λ ′ ∥ ∞ > c ∥ P λ ∥ ∞ / λ with another absolute constant c > 0 . In particular, our construction proves the sharpness of an inequality of Mhaskar for Gaussian networks.
ISSN:0021-9045
1096-0430
DOI:10.1016/j.jat.2006.07.004