Abstract
The arclength of the graphs Γ(S N (f)) of the partial sums S N (f) of the Fourier series of a piecewise C 1 function f with jump discontinuities is equal asymptotically to \((\hbox{the sum of all jumps of $f$})\times L_{N}\), where L N is the Lebesgue constant. This is an improvement of R. Strichartz (J. Fourier Anal. Appl. 6, 533–536, 2000).
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Finch, S.R.: Mathematical Constants. Encyclopedia of Mathematics and Its Applications, vol. 94, Cambridge (2003)
Pinsky, M.A.: Introduction to Fourier Analysis and Wavelets. Graduate Studies in Mathematics, vol. 102. Am. Math. Soc., Providence (2009)
Strichartz, R.S.: Gibbs’ phenomenon and arclength. J. Fourier Anal. Appl. 6, 533–536 (2000)
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Communicated by R. Strichartz.
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Dempo, K., Kuratsubo, S. On Gibbs-Wilbraham Phenomenon and the Arclength of Fourier Series. J Fourier Anal Appl 17, 656–661 (2011). https://doi.org/10.1007/s00041-010-9135-4
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DOI: https://doi.org/10.1007/s00041-010-9135-4