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• Author or Editor: Ulrich Stadtmüller
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# Characterization of the convergence of weighted averages in a more general setting

Studia Scientiarum Mathematicarum Hungarica
Authors: Ferenc Móricz and Ulrich Stadtmüller

Let ν be a positive Borel measure on ℝ̄+:= [0;∞) and let p: ℝ̄+ → ℝ̄+ be a weight function which is locally integrable with respect to ν. We assume that \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $P(t): = \int\limits_0^t {p(u)d\nu (u) \to \infty } andP(t - 0)/P(t) \to 1ast \to \infty .$ \end{document} Let f: ℝ̄+ → ℂ be a locally integrable function with respect to p dν, and define its weighted averages by \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\sigma _t (f;pd\nu ): = \frac{1}{{P(t)}}\int\limits_0^t {f(u)p(u)d\nu (u)}$ \end{document} for large enough t, where P(t) > 0. We prove necessary and sufficient conditions under which the finite limit \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\sigma _t (f;pd\nu ) \to Last \to \infty$ \end{document} exists. This characterization is a unified extension of the results in [5], and it may find application in Probability Theory and Stochastic Processes.

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# Comparison of certain summability methods by speeds of convergence

Analysis Mathematica
Authors: Ulrich Stadtmüller and Anne Tali
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# Characterization of the convergence of weighted averages of sequences and functions

Periodica Mathematica Hungarica
Authors: Ferenc Móricz and Ulrich Stadtmüller

## Abstract

The weighted averages of a sequence (ck), ck ∈ ℂ, with respect to the weights (pk), pk ≥ 0, with {fx135-1} are defined by {fx135-2} while the weighted average of a measurable function f: ℝ+ → ℂ with respect to the weight function p(t) ≥ 0 with {fx135-3}. Under mild assumptions on the weights, we give necessary and sufficient conditions under which the finite limit σnL as n → ∞ or σ(t) → L as t → ∞ exists, respectively. These characterizations may find applications in probability theory.

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