e introduce the remainder
Definition
(Remainder of averaged
Taylor
polynomial)
According to the proposition
(
Properties of averaged
Taylor polynomial
)-2 and formula
(
Mollifier for a ball 2
), for a
function
we
have
We introduce the function
:
According to the proposition
(
Integral form of Taylor
decomposition
),
and
Consequently,
We continue calculation of
:
Our goal is to transform the integral to the form
.We
make the change
in the
-integral,
,
,
We aim to make a change of variables
where
.
Hence, we change the order of integration (see
(
Fubini
theorem
))
and proceed with the change
,
,
,
We would like to put result in the form
,
hence, we reverse the order of integration again. The set of integration is
Note that the line
for a fixed
connects
with a point
in
.
Thus, the
-plane
projection of the set
is the convex hull of
and
:
Let
be the section of
with a plane
:
We continue the
calculation:
where the functions
are given
by
We estimate (see the figure
(
Estimation of averaged
Taylor
polynomial
))
|
Estimation of averaged Taylor
polynomial
|
Thus
We
state
and estimate the
:
Note that
,
hence
By the formulas (
Mollifier for a ball
1
),(
Mollifier for a ball 2
) and the
definition (
Standard mollifier
definition
)-2
where the
depends on
,
so we drop
and substitute for
:
We summarize our findings in the following proposition.
|
