Iterating the Cesàro operators
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- by Fernando Galaz Fontes and Francisco Javier Solís PDF
- Proc. Amer. Math. Soc. 136 (2008), 2147-2153 Request permission
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, $C\!f (x) \ \equiv \ \frac {1}{x} \int _{0}^ {x} f(s) ds$, has two versions: the finite range case is defined for $f \in L^\infty (0,1)$ and the infinite range case for $f \in L^\infty (0, \infty )$. 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, $f(0) = \lim _{x\rightarrow \infty }f(x)$.References
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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
- 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
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 136 (2008), 2147-2153
- MSC (2000): Primary 47B37, 40G05
- DOI: https://doi.org/10.1090/S0002-9939-08-09197-1
- MathSciNet review: 2383520