Editing Pseudo-differential operator (section)

===Linear differential operators with constant coefficients===
Consider a linear [[differential operator]] with constant coefficients,

:<math> P(D) := \sum_\alpha a_\alpha \, D^\alpha </math>

which acts on smooth functions <math>u</math> with compact support in '''R'''<sup>''n''</sup>.
This operator can be written as a composition of a [[Fourier transform]], a simple ''multiplication'' by the
polynomial function (called the '''[[Fourier multiplier|symbol]]''')

:<math>  P(\xi) = \sum_\alpha a_\alpha \, \xi^\alpha, </math>

and an inverse Fourier transform, in the form:

{{NumBlk|:|<math> \quad P(D) u (x) = 
\frac{1}{(2 \pi)^n} \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} e^{i (x - y) \xi} P(\xi) u(y)\, dy \, d\xi </math>|{{EquationRef|1}}}}

Here, <math>\alpha = (\alpha_1,\ldots,\alpha_n)</math> is a [[multi-index]], <math>a_\alpha</math> are complex numbers, and

:<math>D^\alpha=(-i \partial_1)^{\alpha_1} \cdots (-i \partial_n)^{\alpha_n}</math>

is an iterated partial derivative, where ∂<sub>''j''</sub> means differentiation with respect to the ''j''-th variable.  We introduce the constants <math>-i</math> to facilitate the calculation of Fourier transforms.

;Derivation of formula ({{EquationNote|1}})
The Fourier transform of a smooth function ''u'', [[compact support|compactly supported]] in '''R'''<sup>''n''</sup>, is

:<math>\hat u (\xi) := \int e^{- i y \xi} u(y) \, dy</math>

and [[Fourier's inversion formula]] gives

:<math>u (x) = \frac{1}{(2 \pi)^n} \int e^{i x \xi} \hat u (\xi) d\xi = 
\frac{1}{(2 \pi)^n} \iint e^{i (x - y) \xi} u (y) \, dy \, d\xi </math>

By applying ''P''(''D'') to this representation of ''u'' and using

:<math>P(D_x) \, e^{i (x - y) \xi} = e^{i (x - y) \xi} \, P(\xi) </math>

one obtains formula ({{EquationNote|1}}).