Dec 25, 2016 - arXiv:1612.08737v1 [math.FA] 25 Dec 2016. Euler-Maclaurin formulas for functions of bounded variation. Giuseppe De Marcoa, Marco De Zotti, ...
Euler-Maclaurin formulas for functions of bounded variation
arXiv:1612.08737v2 [math.FA] 3 Jan 2017
Giuseppe De Marcoa , Marco De Zotti, Carlo Maricondaa a Dipartimento
di Matematica, Universit` a degli Studi di Padova, Via Trieste 63, 35121 Padova, Italy
Abstract The first-order Euler-Maclaurin formula relates the sum of the values of a smooth function on an interval of integers with its integral on the same interval on R. We formulate here the analogue for functions that are just of bounded variation. Keywords: Euler-Maclaurin, bounded variation, sums, series 2010 MSC: Primary 65B15
Notation Our main reference for the basic facts and related notation on BV functions is [1]. Let us recall that a real valued function f defined on an interval I is of Bounded Variation (we often simply write BV) if the so-called pointwise variation pV(f, I) of f on I, given by X |f (ti+1 ) − f (ti )| : ti ∈ I, t0 < t1 < · · · < tn pV(f, I) := sup 0≤i
