n the section
(
Convolution and smoothing
)
we introduced and stated properties of the standard mollifier
.
We use
to extend Taylor expansion to locally integrable functions.
Note that we do not include
order
derivatives. The reason for it will become apparent when we discuss the
residue term of Taylor decomposition.
To extend the above definition to integrable functions without derivatives we
mollify (see the section
(
Convolution and smoothing
))
over a small ball
and shift all derivatives to the mollifier via integration by parts. Thus, let
be the mollifier over a ball
:


(Mollifier for a ball 1)

where the constant
comes from the
requirement


(Mollifier for a ball 2)

We define the averaged Taylor
polynomial
We introduce the coefficients
by
postulating


(Coefficients a)

and
write
