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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...

Generalizations to monotonicity for uniform convergence of double sine integrals over ℝ̅²₊

Péter Kórus, Ferenc Móricz (2010)

Studia Mathematica

We investigate the convergence behavior of the family of double sine integrals of the form 0 0 f ( x , y ) s i n u x s i n v y d x d y , where (u,v) ∈ ℝ²₊:= ℝ₊ × ℝ₊, ℝ₊:= (0,∞), and f: ℝ²₊ → ℂ is a locally absolutely continuous function satisfying certain generalized monotonicity conditions. We give sufficient conditions for the uniform convergence of the remainder integrals a b a b to zero in (u,v) ∈ ℝ²₊ as maxa₁,a₂ → ∞ and b j > a j 0 , j = 1,2 (called uniform convergence in the regular sense). This implies the uniform convergence of the partial integrals...

Idempotents in quotients and restrictions of Banach algebras of functions

Thomas Vils Pedersen (1996)

Annales de l'institut Fourier

Let 𝒜 β be the Beurling algebra with weight ( 1 + | n | ) β on the unit circle 𝕋 and, for a closed set E 𝕋 , let J 𝒜 β ( E ) = { f 𝒜 β : f = 0 on a neighbourhood of E } . We prove that, for β > 1 2 , there exists a closed set E 𝕋 of measure zero such that the quotient algebra 𝒜 β / J 𝒜 β ( E ) is not generated by its idempotents, thus contrasting a result of Zouakia. Furthermore, for the Lipschitz algebras λ γ and the algebra 𝒜 𝒞 of absolutely continuous functions on 𝕋 , we characterize the closed sets E 𝕋 for which the restriction algebras λ γ ( E ) and 𝒜 𝒞 ( E ) are generated by their idempotents.

Some remarks on descriptive characterizations of the strong McShane integral

Sokol Bush Kaliaj (2019)

Mathematica Bohemica

We present the full descriptive characterizations of the strong McShane integral (or the variational McShane integral) of a Banach space valued function f : W X defined on a non-degenerate closed subinterval W of m in terms of strong absolute continuity or, equivalently, in terms of McShane variational measure V F generated by the primitive F : W X of f , where W is the family of all closed non-degenerate subintervals of W .

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