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

Tauberian conditions for the $$(C, \alpha )$$ ( C , α ) integrability of functions For a real-valued continuous function f(x) on $$[0,\infty )$$ [ 0 , ∞ ) , we define $$\begin{aligned} s(x)=\int _{0}^{x} f(u)du\quad \text {and}\quad \sigma _{\alpha } (x)= \int _{0}^{x}\left( 1-\frac{u}{x}\right) ^{\alpha }f(u)du \end{aligned}$$ s ( x ) = ∫ 0 x f ( u ) d u and σ α ( x ) = ∫ 0 x 1 - u x α f ( u ) d u for $$x>0$$ x > 0 . We say that $$\int _{0}^{\infty } f(u)du$$ ∫ 0 ∞ f ( u ) d u is $$(C, \alpha )$$ ( C , α ) integrable to L for some $$\alpha >-1$$ α > - 1 if the limit $$\lim _{x \rightarrow \infty } \sigma _{\alpha } (x)=L$$ lim x → ∞ σ α ( x ) = L exists. It is known that $$\lim _{x \rightarrow \infty } s(x) =L$$ lim x → ∞ s ( x ) = L implies $$\lim _{x \rightarrow \infty }\sigma _{\alpha } (x) =L$$ lim x → ∞ σ α ( x ) = L for all $$\alpha >-1$$ α > - 1 . The aim of this paper is twofold. First, we introduce some new Tauberian conditions for the $$(C, \alpha )$$ ( C , α ) integrability method under which the converse implication is satisfied, and improve classical Tauberian theorems for the $$(C,\alpha )$$ ( C , α ) integrability method. Next we give short proofs of some classical Tauberian theorems as special cases of some of our results. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

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

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Publisher
Springer International Publishing
Copyright
Copyright © 2016 by Springer International Publishing
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-016-0407-3
Publisher site
See Article on Publisher Site

Abstract

For a real-valued continuous function f(x) on $$[0,\infty )$$ [ 0 , ∞ ) , we define $$\begin{aligned} s(x)=\int _{0}^{x} f(u)du\quad \text {and}\quad \sigma _{\alpha } (x)= \int _{0}^{x}\left( 1-\frac{u}{x}\right) ^{\alpha }f(u)du \end{aligned}$$ s ( x ) = ∫ 0 x f ( u ) d u and σ α ( x ) = ∫ 0 x 1 - u x α f ( u ) d u for $$x>0$$ x > 0 . We say that $$\int _{0}^{\infty } f(u)du$$ ∫ 0 ∞ f ( u ) d u is $$(C, \alpha )$$ ( C , α ) integrable to L for some $$\alpha >-1$$ α > - 1 if the limit $$\lim _{x \rightarrow \infty } \sigma _{\alpha } (x)=L$$ lim x → ∞ σ α ( x ) = L exists. It is known that $$\lim _{x \rightarrow \infty } s(x) =L$$ lim x → ∞ s ( x ) = L implies $$\lim _{x \rightarrow \infty }\sigma _{\alpha } (x) =L$$ lim x → ∞ σ α ( x ) = L for all $$\alpha >-1$$ α > - 1 . The aim of this paper is twofold. First, we introduce some new Tauberian conditions for the $$(C, \alpha )$$ ( C , α ) integrability method under which the converse implication is satisfied, and improve classical Tauberian theorems for the $$(C,\alpha )$$ ( C , α ) integrability method. Next we give short proofs of some classical Tauberian theorems as special cases of some of our results.

Journal

PositivitySpringer Journals

Published: Mar 23, 2016

References

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