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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Iterating the Cesàro operators


Authors: Fernando Galaz Fontes and Francisco Javier Solís
Journal: Proc. Amer. Math. Soc. 136 (2008), 2147-2153
MSC (2000): Primary 47B37, 40G05
Published electronically: February 14, 2008
MathSciNet review: 2383520
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Abstract: The discrete Cesàro operator $ C$ 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 $ s$ 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 $ C^*$, 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 $ f(0)$. 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)$.


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Additional Information

Fernando Galaz Fontes
Affiliation: UAM-Iztapalapa, Av. San Rafael Atlixco 186, Col. Vicentina, México D. F., C. P. 09340
Email: galaz@cimat.mx

Francisco Javier Solís
Affiliation: CIMAT, Apdo. Postal 402, 36 000 Guanajuato, Gto., Mexico
Email: solis@cimat.mx

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09197-1
PII: S 0002-9939(08)09197-1
Keywords: Ces\`aro operator, Iterates, Hausdorff operator
Received by editor(s): November 10, 2006
Received by editor(s) in revised form: April 12, 2007
Published electronically: February 14, 2008
Additional Notes: The first author was partially supported by CONACyT (México) project 49187-F
The second author was partially supported by CONACyT (México) project 50926-F
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.