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

For ergodic averages over $d$-dimensional balls, an integral representation is obtained for $L_2$-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.