  This report aims to present my research updates on distance function wavelets
(DFW) based on the fundamental solutions and the general solutions of the
Helmholtz, modified Helmholtz, and convection-diffusion equations, which
include the isotropic Helmholtz-Fourier (HF) transform and series, the
Helmholtz-Laplace (HL) transform, and the anisotropic convection-diffusion
wavelets and ridgelets. The latter is set to handle discontinuous and track
data problems. The edge effect of the HF series is addressed. Alternative
existence conditions for the DFW transforms are proposed and discussed. To
simplify and streamline the expression of the HF and HL transforms, a new
dimension-dependent function notation is introduced. The HF series is also used
to evaluate the analytical solutions of linear diffusion problems of arbitrary
dimensionality and geometry. The weakness of this report is lacking of rigorous
mathematical analysis due to the author's limited mathematical knowledge.
