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Norms of powers of absolutely convergent fourier series

  • Bogdan M. Baishanski
  • Michael R. Snell
Article
  • 73 Downloads

Abstract

Letf(t) = ∑a k e ikt be infinitely differentiable on R, |f(t)|<1. It is known that under these assumptions ‖n‖ converges to a finite limitl asn → ∞ (l 2 = sec(arga),a = (f′(0))2 -f″(0)). We obtain here more precise results: (i) an asymptotic series (in powers ofn -1/2) for the Fourier coefficientsa nk off n , which holds uniformly ink asn → ∞; (ii) an asymptotic series (this time only powers ofn -1 are present!) for ‖f n ‖; (iii) the fact that ifi j f (j)(0) is real forj = 1,2,..., 2h + 2 then ‖f n ‖ = l + o(n -h ),n → ∞. More generally, we obtain analogous finite asymptotic expansions whenf is assumed to be differentiable only finitely many times.

Keywords

Asymptotic Expansion Fourier Coefficient Central Coefficient Asymptotic Series Taylor Polynomial 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Hebrew University of Jerusalem 1998

Authors and Affiliations

  1. 1.Department of MathematicsThe Ohio State UniversityColumbusUSA

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