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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...

Generalized Analytic and Quasi-Analytic Vectors

Jan Rusinek (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

For every sequence (aₙ) of positive real numbers and an operator acting in a Banach space, we introduce the families of (aₙ)-analytic and (aₙ)-quasi-analytic vectors. We prove various properties of these families.

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