Tauberian theorems for the weighted means of measurable functions of several variables

摘要

Let f, ω: R~n_+ → C and T_ωf(x) denote the weighted mean of f at x with respect to the weight function ω. We prove that the conditions of slow oscillation and slow decrease are Tauberian conditions for the implications: f(x) →st l => f(x) → l and Tωf(x) →st l => f(x) → l. We also prove that the statistical version of the conditions of deferred means are Tauberian conditions for the implication: T_ωf(x) →st l => f(x) →st l. These generalize several well-known results.