n this section we construct an example subspace
and prove the condition (
Energy
approximation
).
We introduce a "partition" set ("mesh")
and define
to be a class of functions
such that
for all sets of numbers
and
.
We define
to be a set of functions from
such
that
The set
is a basis of
.
Indeed, it is linearly independent and for any
we
have
Therefore, for a function
we define an
approximation
and proceed to
estimate
We introduce the convenience
notations
We
estimate
We make a linear change of variables that maps
into
:
thus
We
continue
and introduce the convenience
notation
thus
Note that
by construction of
.
Hence, there is a
such that
then
we apply the formula (
Holder
inequality
)
We substitute the last result into the
estimate
We change the order of
integration
We sum the
estimate
for
and
obtain
