*''linearity''
:$T\_t(a\; f\; +\; b\; h)\; =\; a\; T\_t\; f\; +\; b\; T\_t\; h$
where $f$ and $h$ are signals while $a$ and $b$ are constants,
*''shift invariance''
:$T\_t\; S\_\; f\; =\; S\_\; T\_t\; f$
where $S\_$ denotes the shift (translation) operator $(S\_\; f)(x,\; y)\; =\; f(x-\backslash Delta\; x,\; y\; -\; \backslash Delta\; y)$
*the ''semi-group structure
:$g(x,\; y,\; t\_1)$
* g(x, y, t_2) = g(x, y, t_1 + t_2)
with the associated ''cascade smoothing property''
:$L(x,\; y,\; t\_2)\; =\; g(x,\; y,\; t\_2\; -\; t\_1)$
* L(x, y, t_1)
*existence of an ''infinitesimal generator'' $A$
:$\backslash partial\_t\; L(x,\; y,\; t)\; =\; (A\; L)(x,\; y,\; t)$
*''non-creation of local extrema'' (zero-crossings) in one dimension,
*''non-enhancement of local extrema'' in any number of dimensions
:$\backslash partial\_t\; L(x,\; y,\; t)\; \backslash leq\; 0$ at spatial maxima and $\backslash partial\_t\; L(x,\; y,\; t)\; \backslash geq\; 0$ at spatial minima,
*''rotational symmetry''
:$g(x,\; y,\; t)\; =\; h(x^2+y^2,\; t)$ for some function $h$,
*''scale invariance
:$\backslash hat(\backslash omega\_x,\; \backslash omega\_y,\; t)\; =\; \backslash hat(\backslash frac,\; \backslash frac)$
for some functions $\backslash varphi$ and $\backslash hat$ where $\backslash hat$ denotes the Fourier transform of $g$,
*''positivity'':
:$g(x,\; y,\; t)\; \backslash geq\; 0$,
*''normalization'':
:$\backslash int\_^\; \backslash int\_^\; g(x,\; y,\; t)\; \backslash ,\; dx\; \backslash ,\; dy\; =\; 1$.