The WNA dramatically fails (see Fig. 3.5)
for dilation, translations and rotations
(see Table 3.2 for their definitions in 2D).
It is not surprising that the WNA is `bad' for these
deformations because their
are slowly-changing
delocalized functions of
. However, what is remarkable
is that
for this type of deformations vanishes
in the limit
.
Such deformations we would like to call `special' [14].
Generally, we would like to define a
a deformation as `special' if the associated
fluctuation intensity is
.
A special result that follows from
the considerations of Appendix E
is that a linear combination of special
deformation is also special. Therefore the
special deformations constitute a linear space
of functions. We believe that this linear space
is spanned by the following basis functions:
one dilation, translations, and
rotations.
However we are not able to give a rigorous mathematical
argument that excludes the possibility of having
a larger linear space.
This is discussed in Appendix D (and the generalization
to arbitrary potential
is presented).
In other words, we believe
that any special deformation can be written as
a linear combination of dilation, translations
and rotations.