Editing Differential operator (section)

== Fourier interpretation ==
A differential operator ''P'' and its symbol appear naturally in connection with the [[Fourier transform]] as follows.  Let ƒ be a [[Schwartz function]].  Then by the inverse Fourier transform,

:<math>Pf(x) = \frac{1}{(2\pi)^{\frac{d}{2}}} \int\limits_{\mathbf{R}^d} e^{ ix\cdot\xi} p(x,i\xi)\hat{f}(\xi)\, d\xi.</math>

This exhibits ''P'' as a [[Fourier multiplier]].  A more general class of functions ''p''(''x'',ξ) which satisfy at most polynomial growth conditions in ξ under which this integral is well-behaved comprises the [[pseudo-differential operator]]s.