Differential operator

In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and returns another function (in the style of a higher-order function in computer science).

This article considers mainly linear differential operators, which are the most common type. However, non-linear differential operators also exist, such as the Schwarzian derivative.

Given a nonnegative integer m, an order-${\displaystyle m}$ linear differential operator is a map ${\displaystyle P}$ from a function space ${\displaystyle {\mathcal {F}}_{1}}$ on ${\displaystyle \mathbb {R} ^{n}}$ to another function space ${\displaystyle {\mathcal {F}}_{2}}$ that can be written as:

$${\displaystyle P=\sum _{|\alpha |\leq m}a_{\alpha }(x)D^{\alpha }\ ,}$$ where ${\displaystyle \alpha =(\alpha _{1},\alpha _{2},\cdots ,\alpha _{n})}$ is a multi-index of non-negative integers, ${\displaystyle |\alpha |=\alpha _{1}+\alpha _{2}+\cdots +\alpha _{n}}$, and for each ${\displaystyle \alpha }$, ${\displaystyle a_{\alpha }(x)}$ is a function on some open domain in n-dimensional space. The operator ${\displaystyle D^{\alpha }}$ is interpreted as

$${\displaystyle D^{\alpha }={\frac {\partial ^{|\alpha |}}{\partial x_{1}^{\alpha _{1}}\partial x_{2}^{\alpha _{2}}\cdots \partial x_{n}^{\alpha _{n}}}}}$$

Thus for a function ${\displaystyle f\in {\mathcal {F}}_{1}}$:

$${\displaystyle Pf=\sum _{|\alpha |\leq m}a_{\alpha }(x){\frac {\partial ^{|\alpha |}f}{\partial x_{1}^{\alpha _{1}}\partial x_{2}^{\alpha _{2}}\cdots \partial x_{n}^{\alpha _{n}}}}}$$

The notation ${\displaystyle D^{\alpha }}$ is justified (i.e., independent of order of differentiation) because of the symmetry of second derivatives.