Generalizations to monotonicity for uniform convergence of double sine integrals over $\overline{\mathbb R}^2_+$

Péter Kórus¹ ; Ferenc Móricz¹

¹ Bolyai Institute University of Szeged Aradi vértanúk tere 1 Szeged 6720, Hungary

Studia Mathematica, Tome 201 (2010) no. 3, pp. 287-304

Résumé

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 $f(x,y) \ge 0$.

Détail

DOI : 10.4064/sm201-3-4

Keywords: investigate convergence behavior family double sine integrals form int infty int infty sin sin quad hbox where quad mathbb mathbb times mathbb mathbb infty mathbb mathbb locally absolutely continuous function satisfying certain generalized monotonicity conditions sufficient conditions uniform convergence remainder integrals int int zero mathbb max infty called uniform convergence regular sense implies uniform convergence partial integrals int int mathbb min infty called uniform convergence pringsheims sense these sufficient conditions best possible special