Definition:Cesàro Summation Operator

Definition
Let $\ell^\infty$ be the space of bounded sequences.

Let $\sequence {x_n}_{n \mathop \in \N} \in \ell^\infty$ be a sequence.

Let $A : \ell^\infty \to \ell^\infty$ be an operator such that:


 * $\ds \map A {x_1, x_2, x_3, \ldots} = \tuple {x_1, \frac {x_1 + x_2} 2, \frac {x_1 + x_2 + x_3} 3, \ldots}$

Then $A$ is called the Cesàro summation operator.

Also known as
The Cesàro summation operator is also known as the averaging operator.