Iterating the Cesàro operators

Galaz Fontes, Fernando; Solís, Francisco

doi:10.1090/S0002-9939-08-09197-1

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, $f(0) = \lim _{x\rightarrow \infty }f(x)$.

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