<s>
In	O
the	O
mathematical	O
theory	O
of	O
harmonic	O
analysis	O
,	O
the	O
Riesz	B-Algorithm
transforms	I-Algorithm
are	O
a	O
family	O
of	O
generalizations	O
of	O
the	O
Hilbert	B-Algorithm
transform	I-Algorithm
to	O
Euclidean	O
spaces	O
of	O
dimension	O
d>1	O
.	O
</s>
<s>
They	O
are	O
a	O
type	O
of	O
singular	O
integral	O
operator	O
,	O
meaning	O
that	O
they	O
are	O
given	O
by	O
a	O
convolution	B-Language
of	O
one	O
function	O
with	O
another	O
function	O
having	O
a	O
singularity	O
at	O
the	O
origin	O
.	O
</s>
<s>
The	O
Riesz	B-Algorithm
transforms	I-Algorithm
arises	O
in	O
the	O
study	O
of	O
differentiability	O
properties	O
of	O
harmonic	O
potentials	O
in	O
potential	O
theory	O
and	O
harmonic	O
analysis	O
.	O
</s>
<s>
The	O
Riesz	B-Algorithm
transforms	I-Algorithm
are	O
given	O
by	O
a	O
Fourier	O
multiplier	O
.	O
</s>
<s>
In	O
this	O
form	O
,	O
the	O
Riesz	B-Algorithm
transforms	I-Algorithm
are	O
seen	O
to	O
be	O
generalizations	O
of	O
the	O
Hilbert	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
A	O
particular	O
consequence	O
of	O
this	O
last	O
observation	O
is	O
that	O
the	O
Riesz	B-Algorithm
transform	I-Algorithm
defines	O
a	O
bounded	O
linear	O
operator	O
from	O
L2(Rd )	O
to	O
itself	O
.	O
</s>
<s>
This	O
homogeneity	O
property	O
can	O
also	O
be	O
stated	O
more	O
directly	O
without	O
the	O
aid	O
of	O
the	O
Fourier	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
If	O
σs	O
is	O
the	O
dilation	B-Algorithm
on	O
Rd	O
by	O
the	O
scalar	O
s	O
,	O
that	O
is	O
σsx	O
=	O
sx	O
,	O
then	O
σs	O
defines	O
an	O
action	O
on	O
functions	O
via	O
pullback	O
:	O
</s>
<s>
The	O
Riesz	B-Algorithm
transforms	I-Algorithm
commute	O
with	O
σs	O
:	O
</s>
<s>
Similarly	O
,	O
the	O
Riesz	B-Algorithm
transforms	I-Algorithm
commute	O
with	O
translations	O
.	O
</s>
<s>
For	O
the	O
final	O
property	O
,	O
it	O
is	O
convenient	O
to	O
regard	O
the	O
Riesz	B-Algorithm
transforms	I-Algorithm
as	O
a	O
single	O
vectorial	O
entity	O
Rƒ	O
=(	O
R1ƒ	O
,...,	O
Rdƒ	O
)	O
.	O
</s>
<s>
The	O
final	O
transformation	O
property	O
asserts	O
that	O
the	O
Riesz	B-Algorithm
transform	I-Algorithm
is	O
equivariant	O
with	O
respect	O
to	O
these	O
two	O
actions	O
;	O
that	O
is	O
,	O
</s>
<s>
These	O
three	O
properties	O
in	O
fact	O
characterize	O
the	O
Riesz	B-Algorithm
transform	I-Algorithm
in	O
the	O
following	O
sense	O
.	O
</s>
<s>
T	O
commutes	O
with	O
all	O
dilations	B-Algorithm
and	O
translations	O
.	O
</s>
<s>
Thus	O
the	O
Riesz	B-Algorithm
transform	I-Algorithm
of	O
can	O
be	O
written	O
as	O
:	O
</s>
<s>
so	O
that	O
the	O
Riesz	B-Algorithm
transforms	I-Algorithm
give	O
a	O
way	O
of	O
recovering	O
information	O
about	O
the	O
entire	O
Hessian	O
of	O
a	O
function	O
from	O
knowledge	O
of	O
only	O
its	O
Laplacian	O
.	O
</s>
<s>
Suppose	O
that	O
is	O
a	O
Schwartz	B-Algorithm
function	I-Algorithm
.	O
</s>
