Generalizations to monotonicity for uniform convergence of double sine integrals over $\overline{\mathbb R}^2_+$
Volume 201 / 2010
Abstract
We investigate the convergence behavior of the family of double sine integrals of the form $$\int^\infty_0 \int^\infty_0 f(x,y) \sin ux \sin vy\, dx\, dy, \quad \hbox{where} \quad (u,v) \in \mathbb R^2_+: = \mathbb R_+ \times \mathbb R_+, $$ $\mathbb R_+:= (0, \infty)$, and $f: \mathbb R^2_+ \to \mathbb C$ is a locally absolutely continuous function satisfying certain generalized monotonicity conditions. We give sufficient conditions for the uniform convergence of the remainder integrals $\int^{b_1}_{a_1} \int^{b_2}_{a_2}$ to zero in $(u,v) \in \mathbb R^2_+$ as $\max\{a_1, a_2\} \to \infty$ 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 $\int^{b_1}_0 \int^{b_2}_0$ in $(u,v) \in \mathbb R^2_+$ as $\min \{b_1, b_2\} \to \infty$ (called uniform convergence in Pringsheim's sense). These sufficient conditions are the best possible in the special case when $f(x,y) \ge 0$.
