Wyniki wyszukiwania

1

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

100%

Kórus P., Móricz F.

Studia Mathematica

|

2010

|

tom 201

|

nr 3

287-304

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.

2

On the uniform convergence of double sine series

100%

Kórus P., Móricz F.

Studia Mathematica

|

2009

|

tom 193

|

nr 1

79-97

EN

Let a single sine series (*) $∑^{∞}_{k=1} a_{k} sin kx$ be given with nonnegative coefficients ${a_{k}}$. If ${a_{k}}$ is a "mean value bounded variation sequence" (briefly, MVBVS), then a necessary and sufficient condition for the uniform convergence of series (*) is that $ka_{k} → 0$ as k → ∞. The class MVBVS includes all sequences monotonically decreasing to zero. These results are due to S. P. Zhou, P. Zhou and D. S. Yu. In this paper we extend them from single to double sine series (**) $∑^{∞}_{k=1} ∑^{∞}_{l=1} c_{kl} sin kx sin ly$, even with complex coefficients ${c_{kl}}$. We also give a uniform boundedness test for the rectangular partial sums of series (**), and slightly improve the results on single sine series.

3

Addendum to the paper "Generalizations to monotonicity for uniform convergence of double sine integrals over ℝ̅ ²₊" (Studia Math. 201 (2010), 287-304)

100%

Kórus P., Móricz F.

Studia Mathematica

|

2012

|

tom 212

|

nr 3

285-286

Ograniczanie wyników

3 Studia Mathematica

3 Kórus P.

3 Móricz F.

1 2012

1 2010

1 2009

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

On the uniform convergence of double sine series

Addendum to the paper "Generalizations to monotonicity for uniform convergence of double sine integrals over ℝ̅ ²₊" (Studia Math. 201 (2010), 287-304)