  Helmholtz decomposition theorem for vector fields is usually presented with
too strong restrictions on the fields and only for time independent fields.
Blumenthal showed in 1905 that decomposition is possible for any asymptotically
weakly decreasing vector field. He used a regularization method in his proof
which can be extended to prove the theorem even for vector fields
asymptotically increasing sublinearly. Blumenthal's result is then applied to
the time-dependent fields of the dipole radiation and an artificial sublinearly
increasing field.
