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Iterating the Cesàro operators

Authors:	Fernando Galaz Fontes Francisco Javier Solí s

Institution:	UAM-Iztapalapa, Av. San Rafael Atlixco 186, Col. Vicentina, México D. F., C. P. 09340 ; CIMAT, Apdo. Postal 402, 36 000 Guanajuato, Gto., Mexico

Abstract:	The discrete Cesàro operator associates to a given complex sequence $s = \{s_n\}$ the sequence $Cs \equiv \{b_n \}$ , where $b_n = \frac{s_0 + \dots + s_n}{n +1}, n = 0, 1, \ldots$ . When is a convergent sequence we show that $\{C^n s \}$ converges under the sup-norm if, and only if, $s_0 = \lim_{n\rightarrow\infty} s_n$ . For its adjoint operator , we establish that $\{(C^*)^n s\}$ converges for any $s \in \ell^1$ . The continuous Cesàro operator, $Cf (x) \equiv \frac{1}{x} \int _{0}^ {x}\, f(s) ds$ , has two versions: the finite range case is defined for $f \in L^\infi (0,1)$ and the infinite range case for $f \in L^\infi (0, \infi)$ . In the first situation, when $f: 0, 1] \rightarrow \mathbb{C}$ is continuous we prove that $\{C^n f \}$ converges under the sup-norm to the constant function . In the second situation, when $f: 0, \infty)\rightarrow \mathbb{C}$ is a continuous function having a limit at infinity, we prove that $\{C^n f \}$ converges under the sup-norm if, and only if, $f(0) = \lim_{x\rightarrow \infty}f(x)$ .

Keywords:	Ces\`aro operator Iterates Hausdorff operator

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