<p>For the convergence rates in the ergodic theoremfor unitary actions of locally compact compactly generated abelian groups,we obtain a&#xa0;spectral criterionin terms of the singularity of the spectral measurein a&#xa0;neighborhood of the unit character.It is thus shown thatestimates for these rates,as for the classical ergodic theoremsfor actions of the groups&#xa0;<InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11202_2025_1562_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="9" /> </InlineMediaObject> <EquationSource Format="TEX">$ 𝕑 $</EquationSource> </InlineEquation> and&#xa0;<InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11202_2025_1562_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="9" /> </InlineMediaObject> <EquationSource Format="TEX">$ 𝕉 $</EquationSource> </InlineEquation>studied previously,are necessarily of spectral nature.</p>

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Convergence Rates in the Ergodic Theorem for Unitary Actions of Compactly Generated Abelian Groups

  • A. G. Kachurovskii,
  • I. V. Podvigin,
  • V. È. Todikov

摘要

For the convergence rates in the ergodic theoremfor unitary actions of locally compact compactly generated abelian groups,we obtain a spectral criterionin terms of the singularity of the spectral measurein a neighborhood of the unit character.It is thus shown thatestimates for these rates,as for the classical ergodic theoremsfor actions of the groups  $ 𝕑 $ and  $ 𝕉 $ studied previously,are necessarily of spectral nature.