Tauberian conditions for the $$(C, \alpha )$$ integrability of functions

For a real-valued continuous function f(x) on \([0,\infty )\), we define

$$\begin{aligned} s(x)=\int _{0}^{x} f(u)du\quad \text {and}\quad \sigma _{\alpha } (x)= \int _{0}^{x}\left( 1-\frac{u}{x}\right) ^{\alpha }f(u)du \end{aligned}$$

for \(x>0\). We say that \(\int _{0}^{\infty } f(u)du\) is \((C, \alpha )\) integrable to L for some \(\alpha >-1\) if the limit \(\lim _{x \rightarrow \infty } \sigma _{\alpha } (x)=L\) exists. It is known that \(\lim _{x \rightarrow \infty } s(x) =L\) implies \(\lim _{x \rightarrow \infty }\sigma _{\alpha } (x) =L\) for all \(\alpha >-1\). The aim of this paper is twofold. First, we introduce some new Tauberian conditions for the \((C, \alpha )\) integrability method under which the converse implication is satisfied, and improve classical Tauberian theorems for the \((C,\alpha )\) integrability method. Next we give short proofs of some classical Tauberian theorems as special cases of some of our results.

     

Effect of temperature and membrane preparation parameters on gas permeation properties of polymethacrylates

Permeabilities of N₂, Ar, O₂, CO₂, and H₂ gases in PEMA (Polyethylmethacrylate) membranes have been measured above and below glass transition in the temperature range of 25–70 °C. The permeabilities of the gases were observed increasing with temperature. Arrhenius plot of permeability versus temperature data showed that there is a slope discontinuity at near to T_g of PEMA. In addition, the effects of membrane preparation parameters by solvent casting method (percentage of polymer in solvent, annealing temperature, annealing time, evaporation temperature, and evaporation time) have been investigated by using homogenous dense membranes of PEMA. It is observed that membrane preparation parameters strongly affect the membrane performance and the reproducibility of the permeability measurements. On the other hand, the effect of polymer structure on membrane performance has been investigated. Comparison of the permeabilities of N₂, Ar, O₂, CO₂, and H₂ gases in PEMA and PMMA membranes shows that PMMA membranes have smaller permeabilities and higher selectivities than PEMA membranes because of their higher glass transition temperature, T_g. © 2007 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 45: 3025–3033, 2007     

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