A formulation by Florack 
and continued work by Pauwels et al. 
show that the class of allowable scale-space kernels can be restricted
under weaker conditions, essentially by combining the
conditions about linearity, shift invariance, rotational invariance
and semi-group structure with scale invariance.
It can be shown that for a scale invariant rotationally symmetric semi-group,
the Fourier transform of the convolution kernel must be of the form
for some and p > 0,
which gives a one-parameter class of possible semi-groups.
Florack  proposed to use
separability in Cartesian coordinates as an
additional basic constraint.
Except in the one-dimensional case,
this fixates h to be a Gaussian.
Pauwels et al. showed 
that the corresponding multi-scale representations
have local infinitesimal generators
(basically meaning that the operator in (9)
is a differential operator)
if and only if the exponent p is an even integer.
Out of this countable set of choices,
p = 2 is the only choice that corresponds to
a non-negative convolution kernel
(recall from above that non-creation of
local extrema implies that the kernel has to be non-negative).
Koenderink and van Doorn  carried out
a closely related study,
and showed that derivative operators are natural operators
to derive from a scale-space representation given the assumption
of scale invariance.