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2001/2002 Limit Summability of Real Functions
M. H. Hooshmand Estahbanti
Real Anal. Exchange 27(2): 463-472 (2001/2002).


Let $f$ be a real (or complex) function with domain $D_f$ containing the positive integers. We introduce the functional sequence $\{ f_{\sigma _n}(x) \}$ as follows:$f_{\sigma _n}(x)=xf(n)+\sum_{k=1}^n(f(k)-f(x+k))$ and say that the function $f$ {\sl limit summable} at the point $x_0$ if the sequence $\{ f_{\sigma _n}(x_0) \}$ is convergent, $( f_{\sigma _n}(x_0)\rightarrow f_\sigma (x_0)) \; {\mbox as} \; n \rightarrow \infty$, and we call the function $f_\sigma(x)$ as the limit summand function (of $f$). In this article, we first give a necessary condition for the limit summability of functions and present some elementary properties. Then we prove some tests about limit summability of functions and consider the relation between $f(x)$ and $f_\sigma(x)$. One of the main theorems in this paper gives a uniqueness conditions for a function to be a limit summand function. Finally, as a consequence of this theorem we deduce a generalization of a result due to Bohr-Mollerup.


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M. H. Hooshmand Estahbanti. "Limit Summability of Real Functions." Real Anal. Exchange 27 (2) 463 - 472, 2001/2002.


Published: 2001/2002
First available in Project Euclid: 2 June 2008

zbMATH: 1047.40003
MathSciNet: MR1922662

Primary: 26A99 , 39A10 , 40A30

Keywords: Concave function , concentrable set , convex function , Gamma function , Limit summable function , limit summand function

Rights: Copyright © 2001 Michigan State University Press

Vol.27 • No. 2 • 2001/2002
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