uppose we are facing a generic
problem
where
and
are mappings
for a Hilbert space
and
.
Let
be another Hilbert space and
be an orthogonal
transformation
equivalently
Thus
is any geometrypreserving transformation. In particular, change of basis
fits.
We make the change of unknown
function
Let
for some
.
Then
In context of parabolic PDE,
may be any
transformation
that preserves
geometry.
In particular, Fourier transform in
space
fits as well as decomposition with respect to any orthonormal basis.
For example, let
be a
dependent
basis of
and
so
that
or
We
calculate
Thus, we remove
dependency
from spacial operator if we can find an orthonormal basis
such that
