Summary
The Euler-Maclaurin summation formula and its harmonic analysis (Poisson) are applied to the case of functions which are completely monotone on an open half-line. What thus results is a curious class of Fourier series, which can be determined explicitly and which represent completely monotone functions on the first half of the period. A by-product is the complete monotony (on the first half-period) of the Bernoulli functions, whether the index is integral or fractional.
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Wintner, A. On the reduction (mod 1) of completely monotone functions (0, ∞). Annali di Matematica 43, 299–312 (1957). https://doi.org/10.1007/BF02411911
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DOI: https://doi.org/10.1007/BF02411911