Generalizations to monotonicity for uniform convergence of double sine integrals over (R)over-bar(+)(2)

作者：Peter Korus^*; Ferenc Moricz

来源：Studia Mathematica, 2010, 201(3): 287-304.

DOI：10.4064/sm201-3-4

• convergence of double integrals over (R)over-bar(+)(2) in the regular sense
• convergence in Pringsheim's sense
• monotonically decreasing functions on (R)over-bar(+)(2) mean
• value bounded variation functions (MVBVF)
• non-onesided bounded variation functions (NBVF)
• family of double sine integrals over (R)over-bar(+)(2)
• uniform convergence

We investigate the convergence behavior of the family of double sine integrals of the form
integral(infinity)(0) integral(infinity)(0) (x, y) sin ux sin vy dx dy, where (u, v) is an element of R-+(2) := R+ x R+,
R+ := (0, infinity), and f : R-+(2) -> C is a locally absolutely continuous function satisfying certain generalized monotonicity conditions. We give sufficient conditions for the uniform convergence of the remainder integrals integral(b1)(a1) integral(b2)(a2) to zero in (u, v) is an element of R-+(2) as max{a(1), a(2)} -> infinity 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 integral(b1)(0) integral(b2)(0) (u, v) is an element of R-+(2) as min{b(1), b(2)} -> infinity (called uniform convergence in Pringsheim's sense). These sufficient conditions are the best possible in the special case when f (x, y) >= 0.

• 出版日期2010