<s>
In	O
mathematics	O
,	O
a	O
fusion	B-Algorithm
frame	I-Algorithm
of	O
a	O
vector	O
space	O
is	O
a	O
natural	O
extension	O
of	O
a	O
frame	O
.	O
</s>
<s>
By	O
construction	O
,	O
fusion	B-Algorithm
frames	I-Algorithm
easily	O
lend	O
themselves	O
to	O
parallel	O
or	O
distributed	O
processing	O
of	O
sensor	O
networks	O
consisting	O
of	O
arbitrary	O
overlapping	O
sensor	O
fields	O
.	O
</s>
<s>
When	O
the	O
lower	O
and	O
upper	O
bounds	O
are	O
equal	O
to	O
each	O
other	O
,	O
becomes	O
a	O
-tight	O
fusion	B-Algorithm
frame	I-Algorithm
.	O
</s>
<s>
Furthermore	O
,	O
if	O
,	O
we	O
can	O
call	O
Parseval	O
fusion	B-Algorithm
frame	I-Algorithm
.	O
</s>
<s>
Then	O
is	O
called	O
a	O
fusion	B-Algorithm
frame	I-Algorithm
system	O
for	O
.	O
</s>
<s>
Then	O
is	O
a	O
fusion	B-Algorithm
frame	I-Algorithm
of	O
if	O
and	O
only	O
if	O
is	O
a	O
frame	O
of	O
.	O
</s>
<s>
Additionally	O
,	O
if	O
is	O
called	O
a	O
fusion	B-Algorithm
frame	I-Algorithm
system	O
for	O
with	O
lower	O
and	O
upper	O
bounds	O
and	O
,	O
then	O
is	O
a	O
frame	O
of	O
with	O
lower	O
and	O
upper	O
bounds	O
and	O
.	O
</s>
<s>
And	O
if	O
is	O
a	O
frame	O
of	O
with	O
lower	O
and	O
upper	O
bounds	O
and	O
,	O
then	O
is	O
called	O
a	O
fusion	B-Algorithm
frame	I-Algorithm
system	O
for	O
with	O
lower	O
and	O
upper	O
bounds	O
and	O
.	O
</s>
<s>
Let	O
be	O
a	O
fusion	B-Algorithm
frame	I-Algorithm
for	O
.	O
</s>
<s>
Therefore	O
,	O
the	O
fusion	B-Algorithm
frame	I-Algorithm
operator	O
is	O
positive	O
and	O
invertible	O
.	O
</s>
<s>
For	O
finite	O
frames	O
(	O
i.e.	O
,	O
and	O
)	O
,	O
the	O
fusion	B-Algorithm
frame	I-Algorithm
operator	O
can	O
be	O
constructed	O
with	O
a	O
matrix	O
.	O
</s>
<s>
Let	O
be	O
a	O
fusion	B-Algorithm
frame	I-Algorithm
for	O
,	O
and	O
let	O
be	O
a	O
frame	O
for	O
the	O
subspace	O
and	O
an	O
index	O
set	O
for	O
each	O
.	O
</s>
<s>
The	O
fusion	B-Algorithm
frame	I-Algorithm
operator	O
is	O
then	O
given	O
by	O
an	O
matrix	O
.	O
</s>
