Distribution (mathematics) - Operations On Distributions

Operations On Distributions

Many operations which are defined on smooth functions with compact support can also be defined for distributions. In general, if

is a linear mapping of vector spaces which is continuous with respect to the weak-* topology, then it is possible to extend T to a mapping

by passing to the limit. (This approach works for more general non-linear mappings as well, provided they are assumed to be uniformly continuous.)

In practice, however, it is more convenient to define operations on distributions by means of the transpose (or adjoint transformation) (Strichartz 1994, §2.3; Trèves 1967). If T : D(U) → D(U) is a continuous linear operator, then the transpose is an operator T* : D(U) → D(U) such that

for all φ, ψ ∈ D(U). If such an operator T* exists, and is continuous, then the original operator T may be extended to distributions by defining