Estimates of the remainder in Taylor's theorem using the Henstock--Kurzweil integral

Details Estimates of the remainder in Taylor's theorem using the Henstock--Kurzweil integral Estimates of the remainder in Taylor's theorem using the Henstock--Kurzweil integral

2004-06-18

arxiv.org/abs/math/0406372v1

Mathematics

Classical Analysis and ODEs

To appear in Czechoslovak Mathematical Journal

Scientific paper

When a real-valued function of one variable is approximated by its $n^{th}$ degree Taylor polynomial, the remainder is estimated using the Alexiewicz and Lebesgue $p$-norms in cases where $f^{(n)}$ or $f^{(n+1)}$ are Henstock--Kurzweil integrable. When the only assumption is that $f^{(n)}$ is Henstock--Kurzweil integrable then a modified form of the $n^{th}$ degree Taylor polynomial is used. When the only assumption is that $f^{(n)}\in C^0$ then the remainder is estimated by applying the Alexiewicz norm to Schwartz distributions of order 1.

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