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 geometry-preserving 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
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