Some Tauberian theorems for iterations of Hölder integrability method

Abstract

Let f be a real or complex-valued function on $$[1,\infty )$$ which is continuous over every finite interval [1, x) for $$1<x<\infty $$ . We set $$s(x):=\int _{1}^{x}f(t)dt$$ and define $$\sigma _{k}(s(x))$$ by $$\begin{aligned} \sigma _{k}(s(x))=\left\{ \begin{array}{ll} \displaystyle {\frac{1}{x}\int _{1}^{x}} \sigma _{k-1}(s(t))dt,&{}\quad k\ge 1\\ s(x),&{}\quad k=0 \end{array} \right. \end{aligned}$$ for each nonnegative integer k. An improper integral $$\begin{aligned} \int _{1}^{\infty } f(x)dx \end{aligned}$$ is said to be integrable to a finite number $$\mu $$ by the k-th iteration of Holder or Cesaro mean method of order one, or for short, the (H, k) integrable to $$\mu $$ if $$\begin{aligned} \lim _{x\rightarrow \infty }\sigma _{k}(s(x))=\mu . \end{aligned}$$ In this case, we write $$s(x)\rightarrow \mu \,\,(H,k)$$ . It is clear that the (H, k) integrability method reduces to the ordinary convergence for $$k=0$$ and the (H, 1) integrability method is (C, 1) integrability method. It is known that $$\lim _{x \rightarrow \infty } s(x) =\mu $$ implies $$\lim _{x \rightarrow \infty }\sigma _{k}(s(x)) =\mu $$ . But the converse of this implication is not true in general. In this paper, we obtain some Tauberian conditions for the iterations of Holder integrability method under which the converse implication holds.