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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.

Standard ideals in convolution Sobolev algebras on the half-line

José E. Galé, Antoni Wawrzyńczyk (2011)

Colloquium Mathematicae

We study the relation between standard ideals of the convolution Sobolev algebra ( n ) ( t ) and the convolution Beurling algebra L¹((1+t)ⁿ) on the half-line (0,∞). In particular it is proved that all closed ideals in ( n ) ( t ) with compact and countable hull are standard.

The AP-Denjoy and AP-Henstock integrals revisited

Valentin A. Skvortsov, Piotr Sworowski (2012)

Czechoslovak Mathematical Journal

The note is related to a recently published paper J. M. Park, J. J. Oh, C.-G. Park, D. H. Lee: The AP-Denjoy and AP-Henstock integrals. Czech. Math. J. 57 (2007), 689–696, which concerns a descriptive characterization of the approximate Kurzweil-Henstock integral. We bring to attention known results which are stronger than those contained in the aforementioned work. We show that some of them can be formulated in terms of a derivation basis defined by a local system of which the approximate basis...

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