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On generalized Peano and Peano derivatives

100%

Fejzić H.

Fundamenta Mathematicae

1993

tom 143

nr 1

55-74

A function F is said to have a generalized Peano derivative at x if F is continuous in a neighborhood of x and if there exists a positive integer q such that a qth primitive of F in the neighborhood has the (q+n)th Peano derivative at x; in this case the latter is called the generalized nth Peano derivative of F at x and denoted by $F_{[n]}(x)$. We show that generalized Peano derivatives belong to the class [Δ']. Also we show that they are path derivatives with a nonporous system of paths satisfying the I.I.C. condition as defined in [3]. This gives a new approach to studying generalized Peano and Peano derivatives since all their known properties can be obtained from the corresponding properties of path derivatives. Moreover, generalized Peano derivatives are selective derivatives.

Differentiation of n-convex functions

51%

Fejzić H., Svetic R., Weil C.

Fundamenta Mathematicae

2010

tom 209

nr 1

9-25

The main result of this paper is that if f is n-convex on a measurable subset E of ℝ, then f is n-2 times differentiable, n-2 times Peano differentiable and the corresponding derivatives are equal, and $f^{(n-1)} = f_{(n-1)}$ except on a countable set. Moreover $f_{(n-1)}$ is approximately differentiable with approximate derivative equal to the nth approximate Peano derivative of f almost everywhere.

Ograniczanie wyników

2 Fundamenta Mathematicae

2 Fejzić H.

1 Svetic R.

1 Weil C.

1 2010

1 1993

Wyniki wyszukiwania

On generalized Peano and Peano derivatives

Differentiation of n-convex functions