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

Péter Kórus, and Ferenc Móricz. "Generalizations to monotonicity for uniform convergence of double sine integrals over ℝ̅²₊." Studia Mathematica 201.3 (2010): 287-304. <http://eudml.org/doc/285760>.

@article{PéterKórus2010,
abstract = {We investigate the convergence behavior of the family of double sine integrals of the form $∫_\{0\}^\{∞\} ∫_\{0\}^\{∞\} f(x,y) sin ux sin vy dxdy$, 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.},
author = {Péter Kórus, Ferenc Móricz},
journal = {Studia Mathematica},
keywords = {double sine integral; locally absolutely continuous function; uniform convergence in regular sense; uniform convergence in Pringsheim's sense},
language = {eng},
number = {3},
pages = {287-304},
title = {Generalizations to monotonicity for uniform convergence of double sine integrals over ℝ̅²₊},
url = {http://eudml.org/doc/285760},
volume = {201},
year = {2010},
}

TY - JOUR
AU - Péter Kórus
AU - Ferenc Móricz
TI - Generalizations to monotonicity for uniform convergence of double sine integrals over ℝ̅²₊
JO - Studia Mathematica
PY - 2010
VL - 201
IS - 3
SP - 287
EP - 304
AB - 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.
LA - eng
KW - double sine integral; locally absolutely continuous function; uniform convergence in regular sense; uniform convergence in Pringsheim's sense
UR - http://eudml.org/doc/285760
ER -