Let be an open subset of Euclidean space. One way to state the continuity condition on a distribution on , being a continuous linear functional on the space of bump functions is to require that for all compact subspaces there exist constants and such that the absolute value of is bounded by times the suprema of the sums of the absolute values of all derivatives of of order bounded by :
(order of a distribution)
The distribution has order if serves as a global choice (valid for all compact ) in the above formula, hence if:
If is of order for some , it is said to have finite order.
(e.g. Hoermander 90, def. 2.1.1, below (2.1.2))
Every compactly supported distribution has finite order.
As a special case of example :
A point-supported distribution is a sum of derivatives of the delta distribution at that point. This has order if the order of the derivatives is bounded by .
Last revised on June 11, 2022 at 10:45:43. See the history of this page for a list of all contributions to it.