Iterating the Cesàro operators

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