Definition:Operator of Integrated Weighted Derivatives

Definition
Let $n \in \N$.

Let $\map {a_i} t : \closedint a b \to \R$ be Riemann integrable functions.

Let $h \in \CC^n \closedint a b$ be a Riemann integrable real-valued function of differentiability class $k$.

Then the operator of integrated weighted derivatives is defined as:


 * $\ds \map L h := \int_a^b \sum_{i \mathop = 0}^n \map {a_i} t \map {h^{\paren i}} t \rd t$

where $\ds \int_a^b \map f t \rd r$ denotes the Riemann integral, and $h^{\paren i}$ is the $i$-th derivative of $h$.