Background:
The Helmholtz theorem in vector analysis states that any sufficiently smooth vector field (at least twice differentiable) can be additively split into two components ;
$${\bf v} = \nabla \times {\bf a} + \nabla \phi$$
Where $\bf a$ is a vector field called "vector potential" and $\phi$ is a scalar field called "scalar potential". Now I do not recall any particular definition of the differential operators used.
However, later in my studies I learned about pyramidal algorithms, scale spaces and differential operators defined on such scales. Often within the framework of convolutions and linear operator theory, integral transforms such as the Fourier Transform.
What if we were to define a sequence of differential operators, how will they relate to each other and their correspondingly sufficiently smooth vector fields?
Will each of them be guaranteed their own Helmholtz decomposition with increasingly tougher constraint on smoothness (the vector field $v$ having been low-pass filtered to varying degrees).
Has such a theory been constructed, and if so, how and where can one read about it?