A Tauberian theorem for Cesáro summability of integrals (Q619850)

Suppose \(f\) is a real continuous function and let \(s(x)=\int_{0}^{x}f(t)dt\). The Cesàro mean of \(s(x)\) is given by \(\sigma (s)(x)=x^{-1}\int_{0}^{x}s(t)dt\). If \(s(x)\rightarrow L\), then automatically \(\sigma (s)(x)\rightarrow L\), i.e., \(s(x)\) is Cesàro summable to \(L\). The authors prove that the converse holds if \(s(x)\) is slowly oscillating. As a corollary, it follows that \(\sigma (s)(x)\rightarrow L\) together with the Tauberian condition \(xf(x)=O(1)\) imply that \(s(x)\rightarrow L\).