On the uniform convergence of double sine integrals over –ℝ 2+

Abstract

We investigate the convergence behavior of the family of double sine integrals of the form ∫ ∞ 0 ∫ ∞ 0 f ( x,y ) sin ux sin vy dx dy , where ( u,v ) ∈ ℝ 2 + := ℝ + × ℝ + , ℝ + := (0, ∞), and f is a monotonically nonincreasing function. We give necessary and sufficient conditions for the uniform convergence of the ‘remainder’ integrals ∫ b 1 a 1 ∫ b 2 a 2 to zero in ( u,v ) ∈ ℝ 2 + as max{ a 1 , a 2 } → ∞ , where b j > a j ≥ 0 , j = 1,2 (called uniform convergence in the regular sense). This implies the uniform existence of the finite limits of the partial integrals ∫ b 1 0 ∫ b 2 0 in ( u,v ) ∈ ℝ 2 + as min{ b 1 , b 2 } → ∞ (called uniform convergence in Pringsheim´s sense). Our basic tool is the second mean value theorem for certain double integrals over a rectangle.