# Definition:Differential Operator

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## 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
- $D^\alpha = D_1^{\alpha_1} D_2^{\alpha_2} \dotsm D_n^{\alpha_n}$.