Let \(p:[ 0,\infty)\to( 0,\infty) \) be a continuous function such that the integral \(P( x) =\int_{0}^{x}p( t)\, dt\) is regularly varying of index \(\rho>0,\) that is, \(\lim_{x\to\infty}\frac{P( \mu x)}{P( x)}=\mu^{\rho}\) for all \(\mu>0.\) For a continuous function \(f:[0,\infty) \to\mathbb{R}\) (or \(\mathbb{C}\)), set \(s(x) =\int_{0}^{x}f( t)\, dt\) and \(\sigma_{p}( x)=\frac{1}{P( x)}\int_{0}^{x}s( t) p(t)\, dt.\) The integral \(\int_{0}^{\infty}f( x) dx\) is said to be \(( \overline{N},p) \) summable to a number \(L\) if \(\lim_{x\to\infty}\sigma_{p}( x) =L.\) The summability method \(( \overline{N},p) \) is regular, that is, existence of the integral \(\int_{0}^{\infty}f( x)\, dx\) implies its summability to the same number.
The authors prove some Tauberian theorems related to this method. For example, they show that \[\limsup_{x\to\infty}\int_{x}^{\mu x}\left(\frac{P( t)}{p( t)}\right) ^{r-1}\vert f( t) ^{r}\vert \,dt<\infty\quad ( r,\mu>1) \] is a Tauberian condition for \(( \overline{N},p) \). The authors’ results generalize some classical Tauberian theorems.