et
be a mesh on the interval
,
.
We consider approximation of functions
with piecewise constant functions
,
and investigate possibility to have an estimate of the
form
for some function
.
First, we note that if we require that such estimate would hold for a fixed
mesh and any
then the function
cannot have the desirable
property
Indeed, consider a meshdependent function
:
For the best piecewise constant approximation
we
have
thus
is not possible.
However, it is possible to have
if the mesh
is adapted to a given function
.
We start from
and position
to
satisfy
for some input parameter
.
We continue
recursively
Since
we eventually stop at a step
such that
Let
then
Note
that
thus
is connected to
via a relationship of the
form
