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TAUBERIAN THEOREMS FOR THE WEIGHTED MEANS OF MEASURABLE FUNCTIONS OF SEVERAL VARIABLES

Chang-Pao Chen and Chi-Tung Chang
Taiwanese Journal of Mathematics
Vol. 15, No. 1 (February 2011), pp. 181-199
Stable URL: http://www.jstor.org/stable/taiwjmath.15.1.181
Page Count: 19
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TAUBERIAN THEOREMS FOR THE WEIGHTED MEANS OF MEASURABLE FUNCTIONS OF SEVERAL VARIABLES
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Abstract

Abstract Let f, ω:ℝ+n→ℂ and Tωf (x) denote the weighted mean of f at x with respect to the weight function ω. We prove that the conditions of slow oscillation and slow decrease are Tauberian conditions for the implications: f(x)⟶stl⇒f(x)→l and Tωf(x)⟶stl⇒f(x)→l. We also prove that the statistical version of the conditions of deferred means are Tauberian conditions for the implication: Tωf(x)⟶stl⇒f(x)⟶stl. These generalize several well-known results. 2000 Mathematics Subject Classification: Primary 40A30, 40E05, 40G99. Key words and phrases: Tauberian theorems, Weighted means, Statistical convergence.

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