Real Analysis Exchange

Limit Summability of Real Functions

M. H. Hooshmand Estahbanti

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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.

Article information

Real Anal. Exchange, Volume 27, Number 2 (2001), 463-472.

First available in Project Euclid: 2 June 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 26A99: None of the above, but in this section 40A30: Convergence and divergence of series and sequences of functions 39A10: Difference equations, additive

Limit summable function limit summand function concentrable set convex function concave function Gamma function


Estahbanti, M. H. Hooshmand. Limit Summability of Real Functions. Real Anal. Exchange 27 (2001), no. 2, 463--472.

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