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Absolute continuity with respect to a subset of an interval

Lucie Loukotová (2017)

Commentationes Mathematicae Universitatis Carolinae

The aim of this paper is to introduce a generalization of the classical absolute continuity to a relative case, with respect to a subset M of an interval I . This generalization is based on adding more requirements to disjoint systems { ( a k , b k ) } K from the classical definition of absolute continuity – these systems should be not too far from M and should be small relative to some covers of M . We discuss basic properties of relative absolutely continuous functions and compare this class with other classes of...

Absolutely continuous functions of several variables and diffeomorphisms

Stanislav Hencl, Jan Malý (2003)

Open Mathematics

In [4], a class of absolutely continuous functions of d-variables, motivated by applications to change of variables in an integral, has been introduced. The main result of this paper states that absolutely continuous functions in the sense of [4] are not stable under diffeomorphisms. We also show an example of a function which is absolutely continuous with respect cubes but not with respect to balls.

Absolutely convergent Fourier series and generalized Lipschitz classes of functions

Ferenc Móricz (2008)

Colloquium Mathematicae

We investigate the order of magnitude of the modulus of continuity of a function f with absolutely convergent Fourier series. We give sufficient conditions in terms of the Fourier coefficients in order that f belong to one of the generalized Lipschitz classes Lip(α,L) and Lip(α,1/L), where 0 ≤ α ≤ 1 and L = L(x) is a positive, nondecreasing, slowly varying function such that L(x) → ∞ as x → ∞. For example, a 2π-periodic function f is said to belong to the class Lip(α,L) if | f ( x + h ) - f ( x ) | ≤ C h α L ( 1 / h ) for all x ∈ , h >...

Abstract Perron-Stieltjes integral

Štefan Schwabik (1996)

Mathematica Bohemica

Fundamental results concerning Stieltjes integrals for functions with values in Banach spaces are presented. The background of the theory is the Kurzweil approach to integration, based on Riemann type integral sums (see e.g. [4]). It is known that the Kurzweil theory leads to the (non-absolutely convergent) Perron-Stieltjes integral in the finite dimensional case. In [3] Ch. S. Honig presented a Stieltjes integral for Banach space valued functions. For Honig’s integral the Dushnik interior integral...

Abstract Riemann integrability and measurability

E. de Amo, R. del Campo, M. Díaz Carrillo (2009)

Czechoslovak Mathematical Journal

We prove that the spectral sets of any positive abstract Riemann integrable function are measurable but (at most) a countable amount of them. In addition, the integral of such a function can be computed as an improper classical Riemann integral of the measures of its spectral sets under some weak continuity conditions which in fact characterize the integral representation.

Abstract separation theorems of Rodé type and their applications

Kazimierz Nikodem, Zsolt Páles, Szymon Wąsowicz (1999)

Annales Polonici Mathematici

Sufficient and necessary conditions are presented under which two given functions can be separated by a function Π-affine in Rodé sense (resp. Π-convex, Π-concave). As special cases several old and new separation theorems are obtained.

Adjoint classes of functions in the H 1 sense

Piotr Sworowski (2007)

Czechoslovak Mathematical Journal

Using the concept of the H 1 -integral, we consider a similarly defined Stieltjes integral. We prove a Riemann-Lebesgue type theorem for this integral and give examples of adjoint classes of functions.

Affine and convex functions with respect to the logarithmic mean

Janusz Matkowski (2003)

Colloquium Mathematicae

The class of all functions f:(0,∞) → (0,∞) which are continuous at least at one point and affine with respect to the logarithmic mean is determined. Some related results concerning the functions convex with respect to the logarithmic mean are presented.

Affine Baire functions on Choquet simplices

Miroslav Kačena, Jiří Spurný (2011)

Open Mathematics

We construct a metrizable simplex X such that for each n ɛ ℕ there exists a bounded function f on ext X of Baire class n that cannot be extended to a strongly affine function of Baire class n. We show that such an example cannot be constructed via the space of harmonic functions.

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