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Approximate and L p Peano derivatives of nonintegral order

J. Marshall AshHajrudin Fejzić — 2005

Studia Mathematica

Let n be a nonnegative integer and let u ∈ (n,n+1]. We say that f is u-times Peano bounded in the approximate (resp. L p , 1 ≤ p ≤ ∞) sense at x m if there are numbers f α ( x ) , |α| ≤ n, such that f ( x + h ) - | α | n f α ( x ) h α / α ! is O ( h u ) in the approximate (resp. L p ) sense as h → 0. Suppose f is u-times Peano bounded in either the approximate or L p sense at each point of a bounded measurable set E. Then for every ε > 0 there is a perfect set Π ⊂ E and a smooth function g such that the Lebesgue measure of E∖Π is less than ε and f = g on Π....

Extending Peano derivatives

Hajrudin FejzićJan MaříkClifford E. Weil — 1994

Mathematica Bohemica

Let H [ 0 , 1 ] be a closed set, k a positive integer and f a function defined on H so that the k -th Peano derivative relative to H exists. The major result of this paper is that if H has finite Denjoy index, then f has an extension, F , to [ 0 , 1 ] which is k times Peano differentiable on [ 0 , 1 ] with f i = F i on H for i = 1 , 2 , ... , k .

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