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We establish the localization condition for the γ-means spectral decomposition by the system of fundamental functions of the Laplace operator in an arbitrary multi-dimensional domain. The result is obtained in terms of belonging of decomposing function to the spaces of the generalized Bessel potential. For conditions of localization, we apply the exact estimates for the modulus of continuity of the potential. The generalized Bessel potentials are constructed using convolutions of functions with kernels that generalize the classical Bessel--MacDonald kernels. In contrast to the classical case, non-power singularities of kernels are allowed in the vicinity of the origin. Differential properties of potentials arc described by using the k-th order modulus of continuity in the uniform norm.