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

• Autorzy: Ferenc Móricz
• Języki publikacji: EN

Abstrakty

EN

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 $σ_11 (u,v) := σ(u,v)$

Kategorie tematyczne

• 40B05: Multiple sequences and series (should also be assigned at least one other classification number in this section)
• 40E05: Tauberian theorems, general

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2000
• Tom: 138
• Numer: 1
• Strony: 41-52

Daty

wydano

2000

otrzymano

1998-08-10

poprawiono

1999-11-03

Twórcy

autor

Ferenc Móricz

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

Bibliografia

• [1] G. H. Hardy, Divergent Series, Clarendon Press, Oxford, 1949.
• [2] E. Landau, Über die Bedeutung einiger neuerer Grenzwertsätze der Herren Hardy und Axer, Prace Mat.-Fiz. 21 (1910), 97-177.
• [3] F. Móricz, Tauberian theorems for Cesàro summable double sequences, Studia Math. 110 (1994), 83-96.
• [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] R. Schmidt, Über divergente Folgen und lineare Mittelbindungen, Math. Z. 22 (1925), 89-152.
• [6] E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals, Clarendon Press, Oxford, 1937.