On the power rate of convergence in Wiener’s ergodic theorem

For ergodic averages over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d"> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding="application/x-tex">d</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-dimensional balls, an integral representation is obtained for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L 2"> <mml:semantics> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">L_2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-norms with a kernel containing the Bessel functions of the first kind. Based on this formula, a spectral criterion for the power rate of convergence in Wiener’s ergodic theorem is proved for all possible exponents. The resulting criterion completely covers the known 1-dimensional result.