# Definition:Differential Operator

## Theorem

Let $A$ be a mapping from a function space $\FF_1$ to another function space $\FF_2$.

Let $f \in \FF_2$ be a real function such that $f$ is the image of $u \in \FF_1$ that is:　$f = A \sqbrk u$

A differential operator is represented as a linear combination, finitely generated by $u$ and its derivatives containing higher degree such as

$\displaystyle \map P {x, D} = \sum _{\size \alpha \mathop \le m} \map {a_\alpha} x D^\alpha$

where:

$\alpha = \set {\alpha_1, \alpha_2, \dotsc \alpha_n}$ is a set of non-negative integers forming a multi-index
$\size \alpha = \alpha_1 + \alpha_2 + \dotsb + \alpha_n$ is the length of $\alpha$
the $\map {a_\alpha} x$ are real functions on a open domain in a real cartesian space of $n$ dimensions
$D^\alpha = D_1^{\alpha_1} D_2^{\alpha_2} \dotsm D_n^{\alpha_n}$.