Editing Atiyah–Singer index theorem (section)

===Pseudodifferential operators===
{{main|pseudodifferential operator}}
Pseudodifferential operators can be explained easily in the case of constant coefficient operators on Euclidean space. In this case, constant coefficient differential operators are just the [[Fourier transform]]s of multiplication by polynomials, and constant coefficient pseudodifferential operators are just the Fourier transforms of multiplication by more general functions.

Many proofs of the index theorem use pseudodifferential operators rather than differential operators. The reason for this is that for many purposes there are not enough differential operators. For example, a pseudoinverse of an elliptic differential operator of positive order is not a differential operator, but is a pseudodifferential operator. 
Also, there is a direct correspondence between data representing elements of K(B(''X''), ''S''(''X'')) (clutching functions) and symbols of elliptic pseudodifferential operators.

Pseudodifferential operators have an order, which can be any real number or even −∞, and have symbols (which are no longer polynomials on the cotangent space), and elliptic differential operators are those whose symbols are invertible for sufficiently large cotangent vectors. Most versions of the index theorem can be extended from elliptic differential operators to elliptic pseudodifferential  operators.