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Norms of powers of absolutely convergent fourier series

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Abstract

Letf(t) = ∑a k eikt 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.

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Authors and Affiliations

  1. Department of Mathematics, The Ohio State University, 43210, Columbus, OH, USA

    Bogdan M. Baishanski & Michael R. Snell

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  1. Bogdan M. Baishanski
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  2. Michael R. Snell
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Correspondence to Bogdan M. Baishanski.

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Baishanski, B.M., Snell, M.R. Norms of powers of absolutely convergent fourier series. J. Anal. Math. 76, 137–161 (1998). https://doi.org/10.1007/BF02786933

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