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) ∈ ℝ2+ := 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 ∫b1a1 ∫b2a2 to zero in (u,v) ∈ ℝ2+ as max{a1, a2} → ∞, where bj > aj ≥ 0, j = 1,2 (called uniform convergence in the regular sense). This implies the uniform existence of the finite limits of the partial integrals ∫b10 ∫b20 in (u,v) ∈ ℝ2+ as min{b1, b2} → ∞ (called uniform convergence in Pringsheim´s sense). Our basic tool is the second mean value theorem for certain double integrals over a rectangle.