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Abstract

Schmidt's classical Tauberian theorem says that if a sequence $(s_k : k=0,1,\mathinner {\ldotp \ldotp \ldotp })$ of real numbers is summable $(C,1)$ to a finite limit and slowly decreasing, then it converges to the same limit. In this paper, we prove a nondiscrete version of Schmidt's theorem in the setting of statistical summability $(C,1)$ of real-valued functions that are slowly decreasing on ${{\mathbb R}}_+$. We prove another Tauberian theorem in the case of complex-valued functions that are slowly oscillating on ${{\mathbb R}}_+$. In the proofs we make use of two nondiscrete analogues of the famous Vijayaraghavan lemma, which seem to be new and may be useful in other contexts.