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

Erik Talvila (2005)

Czechoslovak Mathematical Journal

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 ) ∈ C 0 then the remainder is estimated by applying the Alexiewicz norm to Schwartz distributions of order 1.

Examples of functions -extendable for each finite, but not ∞ -extendable

Wiesław Pawłucki (1998)

Banach Center Publications

In Example 1, we describe a subset X of the plane and a function on X which has a k -extension to the whole ℝ 2 for each finite, but has no ∞ -extension to ℝ 2 . In Example 2, we construct a similar example of a subanalytic subset of ℝ 5 ; much more sophisticated than the first one. The dimensions given here are smallest possible.

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