We investigate the convergence behavior of the family of double sine integrals of the form ${\int }_0^{\infty }{\int }_0^{\infty }f(x,y)sinuxsinvydxdy$, 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 ${\int }_{a₁}^{b₁}{\int }_{a₂}^{b₂}$ to zero in (u,v) ∈ ℝ²₊ as maxa₁,a₂ → ∞ and $b_j>a_j\ge 0$, j = 1,2 (called uniform convergence in the regular sense). This implies the uniform convergence of the partial integrals...

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.