Norms of powers of absolutely convergent fourier series

  • Bogdan M. Baishanski
  • Michael R. Snell


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.


Asymptotic Expansion Fourier Coefficient Central Coefficient Asymptotic Series Taylor Polynomial 
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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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