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The operator remainder in the Euler-Maclaurin formula. (English) Zbl 0558.41031
In previous papers the author gave an operator-theoretic form of the Euler-Maclaurin formula, in terms of a derivation D, an anti-derivation and a summation operator, these classes of operators being characterized by the functional equations which they satisfy. That formula contained a power series in odd powers of D. The present paper gives an integral formula for the remainder of the power series after a partial sum. The assumption on D is that \(\sigma:=Sp(D)\setminus \{0\}\) is nonempty and compact, and lies in some open half-plane bounded by a line through 0. The integral formula is a contour integral around \(\sigma\), and uses the Laplace transform of the periodic extension of a Bernoulli polynomial. There is a similar formula for the partial sum iself, using instead the Laplace transform of the Bernoulli polynomial.
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
65B15 Euler-Maclaurin formula in numerical analysis
41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
44A10 Laplace transform
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