## Real Analysis Exchange

### Limit Summability of Real Functions

M. H. Hooshmand Estahbanti

#### Abstract

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

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

Dates
First available in Project Euclid: 2 June 2008

https://projecteuclid.org/euclid.rae/1212412849

Mathematical Reviews number (MathSciNet)
MR1922662

Zentralblatt MATH identifier
1047.40003

#### Citation

Estahbanti, M. H. Hooshmand. Limit Summability of Real Functions. Real Anal. Exchange 27 (2001), no. 2, 463--472. https://projecteuclid.org/euclid.rae/1212412849

#### References

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• R. J. Webster, Log-convex solutions to the functional equation $f(x+1)=g(x)f(x)$: $\Gamma$-type functions, J. Math. Anal. Appl., 209 (1997), 605–623.