# Tauberian theorems for Cesàro summable double integrals over ${ℝ}_{+}^{2}$

Studia Mathematica (2000)

• Volume: 138, Issue: 1, page 41-52
• ISSN: 0039-3223

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## Abstract

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Given ⨍ ∈ ${L}_{l}^{1}oc\left({ℝ}_{+}^{2}\right)$, denote by s(w,z) its integral over the rectangle [0,w]× [0,z] and by σ(u,v) its (C,1,1) mean, that is, the average value of s(w,z) over [0,u] × [0,v], where u,v,w,z>0. Our permanent assumption is that (*) σ(u,v) → A as u,v → ∞, where A is a finite number. First, we consider real-valued functions ⨍ and give one-sided Tauberian conditions which are necessary and sufficient in order that the convergence (**) s(u,v) → A as u,v → ∞ follow from (*). Corollaries allow these Tauberian conditions to be replaced either by Schmidt type slow decrease (or increase) conditions, or by Landau type one-sided Tauberian conditions. Second, we consider complex-valued functions and give a two-sided Tauberian condition which is necessary and sufficient in order that (**) follow from (*). In particular, this condition is satisfied if s(u,v) is slowly oscillating, or if f(x,y) obeys Landau type two-sided Tauberian conditions. At the end, we extend these results to the mixed case, where the (C, 1, 0) mean, that is, the average value of s(w,v) with respect to the first variable over the interval [0,u], is considered instead of ${\sigma }_{1}1\left(u,v\right):=\sigma \left(u,v\right)$

## How to cite

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Móricz, Ferenc. "Tauberian theorems for Cesàro summable double integrals over $ℝ^{2}_{+}$." Studia Mathematica 138.1 (2000): 41-52. <http://eudml.org/doc/216689>.

@article{Móricz2000,
abstract = {Given ⨍ ∈ $L^1_loc (ℝ^2_+)$, denote by s(w,z) its integral over the rectangle [0,w]× [0,z] and by σ(u,v) its (C,1,1) mean, that is, the average value of s(w,z) over [0,u] × [0,v], where u,v,w,z>0. Our permanent assumption is that (*) σ(u,v) → A as u,v → ∞, where A is a finite number. First, we consider real-valued functions ⨍ and give one-sided Tauberian conditions which are necessary and sufficient in order that the convergence (**) s(u,v) → A as u,v → ∞ follow from (*). Corollaries allow these Tauberian conditions to be replaced either by Schmidt type slow decrease (or increase) conditions, or by Landau type one-sided Tauberian conditions. Second, we consider complex-valued functions and give a two-sided Tauberian condition which is necessary and sufficient in order that (**) follow from (*). In particular, this condition is satisfied if s(u,v) is slowly oscillating, or if f(x,y) obeys Landau type two-sided Tauberian conditions. At the end, we extend these results to the mixed case, where the (C, 1, 0) mean, that is, the average value of s(w,v) with respect to the first variable over the interval [0,u], is considered instead of $σ_11 (u,v) := σ(u,v)$},
author = {Móricz, Ferenc},
journal = {Studia Mathematica},
keywords = {Tauberian theorems; Cesàro summable double integrals; Tauberian condition; summability},
language = {eng},
number = {1},
pages = {41-52},
title = {Tauberian theorems for Cesàro summable double integrals over $ℝ^\{2\}_\{+\}$},
url = {http://eudml.org/doc/216689},
volume = {138},
year = {2000},
}

TY - JOUR
AU - Móricz, Ferenc
TI - Tauberian theorems for Cesàro summable double integrals over $ℝ^{2}_{+}$
JO - Studia Mathematica
PY - 2000
VL - 138
IS - 1
SP - 41
EP - 52
AB - Given ⨍ ∈ $L^1_loc (ℝ^2_+)$, denote by s(w,z) its integral over the rectangle [0,w]× [0,z] and by σ(u,v) its (C,1,1) mean, that is, the average value of s(w,z) over [0,u] × [0,v], where u,v,w,z>0. Our permanent assumption is that (*) σ(u,v) → A as u,v → ∞, where A is a finite number. First, we consider real-valued functions ⨍ and give one-sided Tauberian conditions which are necessary and sufficient in order that the convergence (**) s(u,v) → A as u,v → ∞ follow from (*). Corollaries allow these Tauberian conditions to be replaced either by Schmidt type slow decrease (or increase) conditions, or by Landau type one-sided Tauberian conditions. Second, we consider complex-valued functions and give a two-sided Tauberian condition which is necessary and sufficient in order that (**) follow from (*). In particular, this condition is satisfied if s(u,v) is slowly oscillating, or if f(x,y) obeys Landau type two-sided Tauberian conditions. At the end, we extend these results to the mixed case, where the (C, 1, 0) mean, that is, the average value of s(w,v) with respect to the first variable over the interval [0,u], is considered instead of $σ_11 (u,v) := σ(u,v)$
LA - eng
KW - Tauberian theorems; Cesàro summable double integrals; Tauberian condition; summability
UR - http://eudml.org/doc/216689
ER -

## References

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1. [1] G. H. Hardy, Divergent Series, Clarendon Press, Oxford, 1949. Zbl0032.05801
2. [2] E. Landau, Über die Bedeutung einiger neuerer Grenzwertsätze der Herren Hardy und Axer, Prace Mat.-Fiz. 21 (1910), 97-177. Zbl41.0241.01
3. [3] F. Móricz, Tauberian theorems for Cesàro summable double sequences, Studia Math. 110 (1994), 83-96. Zbl0833.40003
4. [4] F. Móricz and Z. Németh, Tauberian conditions under which convergence of integrals follows from summability (C,1) over ${ℝ}_{+}$, Anal. Math. 26 (2000), to appear.
5. [5] R. Schmidt, Über divergente Folgen und lineare Mittelbindungen, Math. Z. 22 (1925), 89-152. Zbl51.0182.04
6. [6] E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals, Clarendon Press, Oxford, 1937. Zbl0017.40404

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