<s>
In	O
mathematics	O
,	O
mollifiers	B-Algorithm
(	O
also	O
known	O
as	O
approximations	O
to	O
the	O
identity	O
)	O
are	O
smooth	O
functions	O
with	O
special	O
properties	O
,	O
used	O
for	O
example	O
in	O
distribution	O
theory	O
to	O
create	O
sequences	O
of	O
smooth	O
functions	O
approximating	O
nonsmooth	O
(	O
generalized	O
)	O
functions	O
,	O
via	O
convolution	B-Language
.	O
</s>
<s>
Intuitively	O
,	O
given	O
a	O
function	O
which	O
is	O
rather	O
irregular	O
,	O
by	O
convolving	O
it	O
with	O
a	O
mollifier	B-Algorithm
the	O
function	O
gets	O
"	O
mollified	O
"	O
,	O
that	O
is	O
,	O
its	O
sharp	O
features	O
are	O
smoothed	O
,	O
while	O
still	O
remaining	O
close	O
to	O
the	O
original	O
nonsmooth	O
(	O
generalized	O
)	O
function	O
.	O
</s>
<s>
They	O
are	O
also	O
known	O
as	O
Friedrichs	B-Algorithm
mollifiers	I-Algorithm
after	O
Kurt	O
Otto	O
Friedrichs	O
,	O
who	O
introduced	O
them	O
.	O
</s>
<s>
Mollifiers	B-Algorithm
were	O
introduced	O
by	O
Kurt	O
Otto	O
Friedrichs	O
in	O
his	O
paper	O
,	O
which	O
is	O
considered	O
a	O
watershed	O
in	O
the	O
modern	O
theory	O
of	O
partial	O
differential	O
equations	O
.	O
</s>
<s>
Flanders	O
was	O
a	O
puritan	O
,	O
nicknamed	O
by	O
his	O
friends	O
Moll	O
after	O
Moll	O
Flanders	O
in	O
recognition	O
of	O
his	O
moral	O
qualities	O
:	O
he	O
suggested	O
to	O
call	O
the	O
new	O
mathematical	O
concept	O
a	O
"	O
mollifier	B-Algorithm
"	O
as	O
a	O
pun	O
incorporating	O
both	O
Flanders	O
 '	O
nickname	O
and	O
the	O
verb	O
'	O
to	O
mollify	O
 '	O
,	O
meaning	O
'	O
to	O
smooth	O
over	O
 '	O
in	O
a	O
figurative	O
sense	O
.	O
</s>
<s>
Previously	O
,	O
Sergei	O
Sobolev	O
used	O
mollifiers	B-Algorithm
in	O
his	O
epoch	O
making	O
1938	O
paper	O
,	O
which	O
contains	O
the	O
proof	O
of	O
the	O
Sobolev	O
embedding	O
theorem	O
:	O
Friedrichs	O
himself	O
acknowledged	O
Sobolev	O
's	O
work	O
on	O
mollifiers	B-Algorithm
stating	O
that:-	O
"	O
These	O
mollifiers	B-Algorithm
were	O
introduced	O
by	O
Sobolev	O
and	O
the	O
author	O
...	O
"	O
.	O
</s>
<s>
It	O
must	O
be	O
pointed	O
out	O
that	O
the	O
term	O
"	O
mollifier	B-Algorithm
"	O
has	O
undergone	O
linguistic	O
drift	O
since	O
the	O
time	O
of	O
these	O
foundational	O
works	O
:	O
Friedrichs	O
defined	O
as	O
"	O
mollifier	B-Algorithm
"	O
the	O
integral	O
operator	O
whose	O
kernel	O
is	O
one	O
of	O
the	O
functions	O
nowadays	O
called	O
mollifiers	B-Algorithm
.	O
</s>
<s>
However	O
,	O
since	O
the	O
properties	O
of	O
a	O
linear	O
integral	O
operator	O
are	O
completely	O
determined	O
by	O
its	O
kernel	O
,	O
the	O
name	O
mollifier	B-Algorithm
was	O
inherited	O
by	O
the	O
kernel	O
itself	O
as	O
a	O
result	O
of	O
common	O
usage	O
.	O
</s>
<s>
where	O
is	O
the	O
Dirac	O
delta	O
function	O
and	O
the	O
limit	O
must	O
be	O
understood	O
in	O
the	O
space	O
of	O
Schwartz	O
distributions	O
,	O
then	O
is	O
a	O
mollifier	B-Algorithm
.	O
</s>
<s>
When	O
the	O
theory	O
of	O
distributions	O
was	O
still	O
not	O
widely	O
known	O
nor	O
used	O
,	O
property	O
above	O
was	O
formulated	O
by	O
saying	O
that	O
the	O
convolution	B-Language
of	O
the	O
function	O
with	O
a	O
given	O
function	O
belonging	O
to	O
a	O
proper	O
Hilbert	O
or	O
Banach	O
space	O
converges	O
as	O
ε	O
→	O
0	O
to	O
that	O
function	O
:	O
this	O
is	O
exactly	O
what	O
Friedrichs	O
did	O
.	O
</s>
<s>
This	O
also	O
clarifies	O
why	O
mollifiers	B-Algorithm
are	O
related	O
to	O
approximate	O
identities	O
.	O
</s>
<s>
As	O
briefly	O
pointed	O
out	O
in	O
the	O
"	O
Historical	O
notes	O
"	O
section	O
of	O
this	O
entry	O
,	O
originally	O
,	O
the	O
term	O
"	O
mollifier	B-Algorithm
"	O
identified	O
the	O
following	O
convolution	B-Language
operator	I-Language
:	O
</s>
<s>
This	O
function	O
is	O
infinitely	O
differentiable	O
,	O
non	O
analytic	O
with	O
vanishing	O
derivative	B-Algorithm
for	O
.	O
</s>
<s>
can	O
be	O
therefore	O
used	O
as	O
mollifier	B-Algorithm
as	O
described	O
above	O
:	O
one	O
can	O
see	O
that	O
defines	O
a	O
positive	O
and	O
symmetric	O
mollifier	B-Algorithm
.	O
</s>
<s>
All	O
properties	O
of	O
a	O
mollifier	B-Algorithm
are	O
related	O
to	O
its	O
behaviour	O
under	O
the	O
operation	O
of	O
convolution	B-Language
:	O
we	O
list	O
the	O
following	O
ones	O
,	O
whose	O
proofs	O
can	O
be	O
found	O
in	O
every	O
text	O
on	O
distribution	O
theory	O
.	O
</s>
<s>
where	O
denotes	O
convolution	B-Language
,	O
is	O
a	O
family	O
of	O
smooth	O
functions	O
.	O
</s>
<s>
where	O
indicates	O
the	O
support	O
in	O
the	O
sense	O
of	O
distributions	O
,	O
and	O
indicates	O
their	O
Minkowski	B-Algorithm
addition	I-Algorithm
.	O
</s>
<s>
The	O
basic	O
application	O
of	O
mollifiers	B-Algorithm
is	O
to	O
prove	O
that	O
properties	O
valid	O
for	O
smooth	O
functions	O
are	O
also	O
valid	O
in	O
nonsmooth	O
situations	O
:	O
</s>
<s>
Very	O
informally	O
,	O
mollifiers	B-Algorithm
are	O
used	O
to	O
prove	O
the	O
identity	O
of	O
two	O
different	O
kind	O
of	O
extension	O
of	O
differential	O
operators	O
:	O
the	O
strong	O
extension	O
and	O
the	O
weak	B-Algorithm
extension	I-Algorithm
.	O
</s>
