Cauchy Principal Value - Distribution Theory

Distribution Theory

Let be the set of smooth functions with compact support on the real line Then, the map

defined via the Cauchy principal value as

is a distribution. The map itself may sometimes be called the principal value (hence the notation p.v.). This distribution appears for example in the Fourier transform of the Heaviside step function.

The principal value is not exclusively defined on smooth functions ; it is enough that u be integrable, with compact support and differentiable at point 0.

It is the inverse distribution of function x and is almost the only distribution with this property :

where K is a constant and δ the Dirac distribution.

More generally, the principal value can be defined for a wide class of singular integral kernels on the Euclidean space Rn. If K(x) has an isolated singularity at the origin, but is an otherwise "nice" function, then the principal value distribution is defined on compactly supported smooth functions by

Such a limit may not be well defined or, being well-defined, it may not necessarily define a distribution. It is, however, well-defined if K is a continuous homogeneous function of degree −n whose integral over any sphere centered at the origin vanishes. This is the case, for instance, with the Riesz transforms.