General logarithmic control modulo and Tauberian remainder theorems

Abstract

Let $\lambda=(\lambda_n)$ be a nondecreasing sequence of positive numbers such that $\lambda_n\to\infty$. A sequence $(\xi_n)$ is called $\lambda$-bounded if \begin{equation*} \lambda_n(\xi_n-\alpha)=O(1)\end{equation*} with the limit $\displaystyle{\lim_{n\rightarrow \infty}\xi_n=\alpha}$. In this work, we obtain several Tauberian remainder theorems on $\lambda$-bounded sequences for the logarithmic summability method with help of general logarithmic control modulo of the oscillatory behavior. Tauber conditions in our main results are on the generator sequence and the general logarithmic control modulo.