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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.
MSC:
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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References:
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