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Studia Scientiarum Mathematicarum Hungarica
Author:
Árpád Fekete
The notions of statistical limit, limit inferior and limit superior of a measurable function at
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\(\infty\)
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were introduced by Móricz. These notions can be considered as the nondiscrete analogues of those introduced for sequences of numbers by H. Fast, J. A. Fridy and C. Orhan. Let
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\(0 \not \equiv p\: \mathbb{R}_+ \to \mathbb{R}_+\)
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be a nondecreasing function such that
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\(p(0)=0\)
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and \documentclass{aastex}
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$$\mbox{st-\!}\liminf_{t \to \infty} \frac{p(\lambda t)}{p(t)} >1 \ \text{for every} \lambda >1.$$
\end{document} Given a real- or complex-valued function
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\(f \in L_{{\rm loc}}^1 (\mathbb{R}_+)\)
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, we define \documentclass{aastex}
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$$s(x):= \int^x_0 f(u) \, du\ \text{and}\ \sigma(t) := \frac{1}{p(t)} \int^t_0 s(x) d p(x),\quad t>0.$$
\end{document} Our goal is to find necessary and sufficient conditions under which the existence of
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\(\mbox{st-}\lim s(t)=l\)
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follows from that of
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\(\mbox{st-}\lim \sigma(t)=l\)
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, where
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\(l\)
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is a finite number. In the case of real-valued functions we present one-sided Tauberian conditions, while in the case of complex-valued functions we present two-sided Tauberian conditions.
