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

• Autorzy: Péter Kórus , Ferenc Móricz
• Języki publikacji: EN

Abstrakty

EN

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 max{a₁,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 min{b₁,b₂} → ∞ (called uniform convergence in Pringsheim's sense). These sufficient conditions are the best possible in the special case when f(x,y) ≥ 0.

Kategorie tematyczne

• 26A46: Absolutely continuous functions
• 42B99: None of the above, but in this section
• 26B30: Absolutely continuous functions, functions of bounded variation
• 42A32: Trigonometric series of special types (positive coefficients, monotonic coefficients, etc.)
• 40A10: Convergence and divergence of integrals

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2010
• Tom: 201
• Numer: 3
• Strony: 287-304

Daty

wydano

2010

Twórcy

autor

Péter Kórus

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

autor

Ferenc Móricz

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

Identyfikatory

DOI

10.4064/sm201-3-4