Generalizations to monotonicity for uniform convergence of double sine integrals over ℝ̅²₊

Péter Kórus ; Ferenc Móricz

Studia Mathematica, Tome 196 (2010), p. 287-304 / Harvested from The Polish Digital Mathematics Library

Résumé

We investigate the convergence behavior of the family of double sine integrals of the form $∫_{0}^{∞} ∫_{0}^{∞} f(x,y) sin ux sin vy dxdy$, 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 $∫^{b₁}_{a₁} ∫^{b₂}_{a₂}$ to zero in (u,v) ∈ ℝ²₊ as maxa₁,a₂ → ∞ and $b_{j} > a_{j} ≥ 0$, j = 1,2 (called uniform convergence in the regular sense). This implies the uniform convergence of the partial integrals $∫_{0}^{b₁} ∫_{0}^{b₂}$ in (u,v) ∈ ℝ²₊ as minb₁,b₂ → ∞ (called uniform convergence in Pringsheim’s sense). These sufficient conditions are the best possible in the special case when f(x,y) ≥ 0.

Détail

Publié le : 2010-01-01

Zbl 1215.26009

EUDML-ID : urn:eudml:doc:285760