<s>
In	O
mathematics	O
,	O
the	O
Weierstrass	B-Algorithm
transform	I-Algorithm
of	O
a	O
function	O
,	O
named	O
after	O
Karl	O
Weierstrass	O
,	O
is	O
a	O
"	O
smoothed	O
"	O
version	O
of	O
obtained	O
by	O
averaging	O
the	O
values	O
of	O
,	O
weighted	O
with	O
a	O
Gaussian	O
centered	O
atx	O
.	O
</s>
<s>
The	O
factor	O
1/	O
√( 4π	O
)	O
is	O
chosen	O
so	O
that	O
the	O
Gaussian	O
will	O
have	O
a	O
total	O
integral	O
of	O
1	O
,	O
with	O
the	O
consequence	O
that	O
constant	O
functions	O
are	O
not	O
changed	O
by	O
the	O
Weierstrass	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
The	O
Weierstrass	B-Algorithm
transform	I-Algorithm
is	O
intimately	O
related	O
to	O
the	O
heat	O
equation	O
(	O
or	O
,	O
equivalently	O
,	O
the	O
diffusion	O
equation	O
with	O
constant	O
diffusion	O
coefficient	O
)	O
.	O
</s>
<s>
If	O
the	O
function	O
describes	O
the	O
initial	O
temperature	O
at	O
each	O
point	O
of	O
an	O
infinitely	O
long	O
rod	O
that	O
has	O
constant	O
thermal	O
conductivity	O
equal	O
to	O
1	O
,	O
then	O
the	O
temperature	O
distribution	O
of	O
the	O
rod	O
t	O
=	O
1	O
time	O
units	O
later	O
will	O
be	O
given	O
by	O
the	O
function	O
F	O
.	O
By	O
using	O
values	O
of	O
t	O
different	O
from	O
1	O
,	O
we	O
can	O
define	O
the	O
generalized	O
Weierstrass	B-Algorithm
transform	I-Algorithm
of	O
.	O
</s>
<s>
The	O
generalized	O
Weierstrass	B-Algorithm
transform	I-Algorithm
provides	O
a	O
means	O
to	O
approximate	O
a	O
given	O
integrable	O
function	O
arbitrarily	O
well	O
with	O
analytic	B-Language
functions	I-Language
.	O
</s>
<s>
It	O
is	O
also	O
known	O
as	O
the	O
Gauss	B-Algorithm
transform	I-Algorithm
or	O
Gauss	O
–	O
Weierstrass	B-Algorithm
transform	I-Algorithm
after	O
Carl	O
Friedrich	O
Gauss	O
and	O
as	O
the	O
Hille	O
transform	O
after	O
Einar	O
Carl	O
Hille	O
who	O
studied	O
it	O
extensively	O
.	O
</s>
<s>
The	O
generalization	O
Wt	O
mentioned	O
below	O
is	O
known	O
in	O
signal	O
analysis	O
as	O
a	O
Gaussian	O
filter	O
and	O
in	O
image	B-Algorithm
processing	I-Algorithm
(	O
when	O
implemented	O
on	O
R2	O
)	O
as	O
a	O
Gaussian	B-Error_Name
blur	I-Error_Name
.	O
</s>
<s>
As	O
mentioned	O
above	O
,	O
every	O
constant	O
function	O
is	O
its	O
own	O
Weierstrass	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
The	O
Weierstrass	B-Algorithm
transform	I-Algorithm
of	O
any	O
polynomial	O
is	O
a	O
polynomial	O
of	O
the	O
same	O
degree	O
,	O
and	O
in	O
fact	O
same	O
leading	O
coefficient	O
(	O
the	O
asymptotic	O
growth	O
is	O
unchanged	O
)	O
.	O
</s>
<s>
Indeed	O
,	O
if	O
denotes	O
the	O
(	O
physicist	O
's	O
)	O
Hermite	O
polynomial	O
of	O
degree	O
n	O
,	O
then	O
the	O
Weierstrass	B-Algorithm
transform	I-Algorithm
of	O
(	O
/2	O
)	O
is	O
simply	O
.	O
</s>
<s>
This	O
can	O
be	O
shown	O
by	O
exploiting	O
the	O
fact	O
that	O
the	O
generating	O
function	O
for	O
the	O
Hermite	O
polynomials	O
is	O
closely	O
related	O
to	O
the	O
Gaussian	O
kernel	O
used	O
in	O
the	O
definition	O
of	O
the	O
Weierstrass	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
The	O
Weierstrass	B-Algorithm
transform	I-Algorithm
of	O
the	O
function	O
eax	O
(	O
where	O
a	O
is	O
an	O
arbitrary	O
constant	O
)	O
is	O
ea2eax	O
.	O
</s>
<s>
The	O
function	O
eax	O
is	O
thus	O
an	O
eigenfunction	B-Algorithm
of	O
the	O
Weierstrass	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
(	O
This	O
is	O
,	O
in	O
fact	O
,	O
more	O
generally	O
true	O
for	O
all	O
convolution	B-Language
transforms	O
.	O
)	O
</s>
<s>
Setting	O
a	O
=	O
bi	O
where	O
i	O
is	O
the	O
imaginary	O
unit	O
,	O
and	O
applying	O
Euler	O
's	O
identity	O
,	O
one	O
sees	O
that	O
the	O
Weierstrass	B-Algorithm
transform	I-Algorithm
of	O
the	O
function	O
cos(bx )	O
is	O
e−	O
b2cos(bx )	O
and	O
the	O
Weierstrass	B-Algorithm
transform	I-Algorithm
of	O
the	O
function	O
sin(bx )	O
is	O
e−	O
b2sin(bx )	O
.	O
</s>
<s>
In	O
particular	O
,	O
by	O
choosing	O
a	O
negative	O
,	O
it	O
is	O
evident	O
that	O
the	O
Weierstrass	B-Algorithm
transform	I-Algorithm
of	O
a	O
Gaussian	O
function	O
is	O
again	O
a	O
Gaussian	O
function	O
,	O
but	O
a	O
"	O
wider	O
"	O
one	O
.	O
</s>
<s>
The	O
Weierstrass	B-Algorithm
transform	I-Algorithm
assigns	O
to	O
each	O
function	O
f	O
a	O
new	O
function	O
F	O
;	O
this	O
assignment	O
is	O
linear	B-Architecture
.	O
</s>
<s>
Both	O
of	O
these	O
facts	O
are	O
more	O
generally	O
true	O
for	O
any	O
integral	O
transform	O
defined	O
via	O
convolution	B-Language
.	O
</s>
<s>
If	O
the	O
transform	O
F(x )	O
exists	O
for	O
the	O
real	O
numbers	O
x	O
=	O
a	O
and	O
x	O
=	O
b	O
,	O
then	O
it	O
also	O
exists	O
for	O
all	O
real	O
values	O
in	O
between	O
and	O
forms	O
an	O
analytic	B-Language
function	I-Language
there	O
;	O
moreover	O
,	O
F(x )	O
will	O
exist	O
for	O
all	O
complex	O
values	O
of	O
x	O
with	O
a≤	O
Re(x )	O
≤	O
b	O
and	O
forms	O
a	O
holomorphic	O
function	O
on	O
that	O
strip	O
of	O
the	O
complex	O
plane	O
.	O
</s>
<s>
f∈	O
L1(R )	O
)	O
,	O
then	O
so	O
is	O
its	O
Weierstrass	B-Algorithm
transform	I-Algorithm
F	O
,	O
and	O
if	O
furthermore	O
f(x )	O
≥0	O
for	O
all	O
x	O
,	O
then	O
also	O
F(x )	O
≥0	O
for	O
all	O
x	O
and	O
the	O
integrals	O
of	O
f	O
and	O
F	O
are	O
equal	O
.	O
</s>
<s>
The	O
Weierstrass	B-Algorithm
transform	I-Algorithm
consequently	O
yields	O
a	O
bounded	O
operator	O
W	O
:	O
Lp(R )	O
→	O
Lp(R )	O
.	O
</s>
<s>
If	O
f	O
is	O
sufficiently	O
smooth	O
,	O
then	O
the	O
Weierstrass	B-Algorithm
transform	I-Algorithm
of	O
the	O
k-th	O
derivative	B-Algorithm
of	O
f	O
is	O
equal	O
to	O
the	O
k-th	O
derivative	B-Algorithm
of	O
the	O
Weierstrass	B-Algorithm
transform	I-Algorithm
off	O
.	O
</s>
<s>
We	O
have	O
seen	O
above	O
that	O
the	O
Weierstrass	B-Algorithm
transform	I-Algorithm
of	O
cos(bx )	O
is	O
e−b2	O
cos(bx )	O
,	O
and	O
analogously	O
for	O
sin(bx )	O
.	O
</s>
<s>
contains	O
a	O
summand	O
which	O
is	O
a	O
combination	O
of	O
sin(bx )	O
and	O
cos(bx )	O
)	O
,	O
then	O
the	O
transformed	O
signal	O
F	O
will	O
contain	O
the	O
same	O
frequency	O
,	O
but	O
with	O
an	O
amplitude	B-Application
multiplied	O
by	O
the	O
factor	O
e−b2	O
.	O
</s>
<s>
This	O
has	O
the	O
consequence	O
that	O
higher	O
frequencies	O
are	O
reduced	O
more	O
than	O
lower	O
ones	O
,	O
and	O
the	O
Weierstrass	B-Algorithm
transform	I-Algorithm
thus	O
acts	O
as	O
a	O
low-pass	B-Algorithm
filter	I-Algorithm
.	O
</s>
<s>
This	O
can	O
also	O
be	O
shown	O
with	O
the	O
continuous	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
,	O
as	O
follows	O
.	O
</s>
<s>
The	O
Fourier	B-Algorithm
transform	I-Algorithm
analyzes	O
a	O
signal	O
in	O
terms	O
of	O
its	O
frequencies	O
,	O
transforms	O
convolutions	B-Language
into	O
products	O
,	O
and	O
transforms	O
Gaussians	O
into	O
Gaussians	O
.	O
</s>
<s>
The	O
Weierstrass	B-Algorithm
transform	I-Algorithm
is	O
convolution	B-Language
with	O
a	O
Gaussian	O
and	O
is	O
therefore	O
multiplication	O
of	O
the	O
Fourier	O
transformed	O
signal	O
with	O
a	O
Gaussian	O
,	O
followed	O
by	O
application	O
of	O
the	O
inverse	O
Fourier	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
This	O
multiplication	O
with	O
a	O
Gaussian	O
in	O
frequency	O
space	O
blends	O
out	O
high	O
frequencies	O
,	O
which	O
is	O
another	O
way	O
of	O
describing	O
the	O
"	O
smoothing	O
"	O
property	O
of	O
the	O
Weierstrass	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
The	O
following	O
formula	O
,	O
closely	O
related	O
to	O
the	O
Laplace	O
transform	O
of	O
a	O
Gaussian	O
function	O
,	O
and	O
a	O
real	O
analogue	O
to	O
the	O
Hubbard	B-Algorithm
–	I-Algorithm
Stratonovich	I-Algorithm
transformation	I-Algorithm
,	O
is	O
relatively	O
easy	O
to	O
establish	O
:	O
</s>
<s>
to	O
thus	O
obtain	O
the	O
following	O
formal	O
expression	O
for	O
the	O
Weierstrass	B-Algorithm
transform	I-Algorithm
W	O
,	O
</s>
<s>
The	O
above	O
formal	O
derivation	B-Algorithm
glosses	O
over	O
details	O
of	O
convergence	O
,	O
and	O
the	O
formula	O
W	O
=	O
eD2	O
is	O
thus	O
not	O
universally	O
valid	O
;	O
there	O
are	O
several	O
functions	O
f	O
which	O
have	O
a	O
well-defined	O
Weierstrass	B-Algorithm
transform	I-Algorithm
,	O
but	O
for	O
which	O
eD2f(x )	O
cannot	O
be	O
meaningfully	O
defined	O
.	O
</s>
<s>
Nevertheless	O
,	O
the	O
rule	O
is	O
still	O
quite	O
useful	O
and	O
can	O
,	O
for	O
example	O
,	O
be	O
used	O
to	O
derive	O
the	O
Weierstrass	B-Algorithm
transforms	I-Algorithm
of	O
polynomials	O
,	O
exponential	O
and	O
trigonometric	O
functions	O
mentioned	O
above	O
.	O
</s>
<s>
But	O
if	O
,	O
for	O
instance	O
,	O
f∈	O
L2(R )	O
,	O
then	O
knowledge	O
of	O
all	O
the	O
derivatives	B-Algorithm
of	O
F	O
at	O
x	O
=	O
0	O
suffices	O
to	O
yield	O
the	O
coefficients	O
an	O
;	O
and	O
to	O
thus	O
reconstruct	O
as	O
a	O
series	O
of	O
Hermite	O
polynomials	O
.	O
</s>
<s>
A	O
third	O
method	O
of	O
inverting	O
the	O
Weierstrass	B-Algorithm
transform	I-Algorithm
exploits	O
its	O
connection	O
to	O
the	O
Laplace	O
transform	O
mentioned	O
above	O
,	O
and	O
the	O
well-known	O
inversion	O
formula	O
for	O
the	O
Laplace	O
transform	O
.	O
</s>
<s>
We	O
can	O
use	O
convolution	B-Language
with	O
the	O
Gaussian	O
kernel	O
(	O
with	O
some	O
)	O
instead	O
of	O
,	O
thus	O
defining	O
an	O
operator	O
,	O
the	O
generalized	O
Weierstrass	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
convolution	B-Language
with	O
the	O
Dirac	O
delta	O
function	O
)	O
,	O
and	O
these	O
then	O
form	O
a	O
one-parameter	O
semigroup	O
of	O
operators	O
.	O
</s>
<s>
The	O
kernel	O
used	O
for	O
the	O
generalized	O
Weierstrass	B-Algorithm
transform	I-Algorithm
is	O
sometimes	O
called	O
the	O
Gauss	O
–	O
Weierstrass	O
kernel	O
,	O
and	O
is	O
Green	O
's	O
function	O
for	O
the	O
diffusion	O
equation	O
on	O
.	O
</s>
<s>
The	O
Weierstrass	B-Algorithm
transform	I-Algorithm
can	O
also	O
be	O
defined	O
for	O
certain	O
classes	O
of	O
distributions	O
or	O
"	O
generalized	O
functions	O
"	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
Weierstrass	B-Algorithm
transform	I-Algorithm
of	O
the	O
Dirac	O
delta	O
is	O
the	O
Gaussian	O
.	O
</s>
<s>
Furthermore	O
,	O
the	O
Weierstrass	B-Algorithm
transform	I-Algorithm
can	O
be	O
defined	O
for	O
real	O
-	O
(	O
or	O
complex	O
-	O
)	O
valued	O
functions	O
(	O
or	O
distributions	O
)	O
defined	O
on	O
.	O
</s>
<s>
We	O
use	O
the	O
same	O
convolution	B-Language
formula	O
as	O
above	O
but	O
interpret	O
the	O
integral	O
as	O
extending	O
over	O
all	O
of	O
and	O
the	O
expression	O
as	O
the	O
square	O
of	O
the	O
Euclidean	O
length	O
of	O
the	O
vector	O
;	O
the	O
factor	O
in	O
front	O
of	O
the	O
integral	O
has	O
to	O
be	O
adjusted	O
so	O
that	O
the	O
Gaussian	O
will	O
have	O
a	O
total	O
integral	O
of1	O
.	O
</s>
<s>
More	O
generally	O
,	O
the	O
Weierstrass	B-Algorithm
transform	I-Algorithm
can	O
be	O
defined	O
on	O
any	O
Riemannian	B-Architecture
manifold	I-Architecture
:	O
the	O
heat	O
equation	O
can	O
be	O
formulated	O
there	O
(	O
using	O
the	O
manifold	O
's	O
Laplace	O
–	O
Beltrami	O
operator	O
)	O
,	O
and	O
the	O
Weierstrass	B-Algorithm
transform	I-Algorithm
is	O
then	O
given	O
by	O
following	O
the	O
solution	O
of	O
the	O
heat	O
equation	O
for	O
one	O
time	O
unit	O
,	O
starting	O
with	O
the	O
initial	O
"	O
temperature	O
distribution	O
"	O
.	O
</s>
<s>
If	O
one	O
considers	O
convolution	B-Language
with	O
the	O
kernel	O
instead	O
of	O
with	O
a	O
Gaussian	O
,	O
one	O
obtains	O
the	O
Poisson	O
transform	O
which	O
smoothes	O
and	O
averages	O
a	O
given	O
function	O
in	O
a	O
manner	O
similar	O
to	O
the	O
Weierstrass	B-Algorithm
transform	I-Algorithm
.	O
</s>
