<s>
In	O
applied	O
mathematics	O
,	O
the	O
transfer	B-Algorithm
matrix	I-Algorithm
is	O
a	O
formulation	O
in	O
terms	O
of	O
a	O
block-Toeplitz	B-Algorithm
matrix	I-Algorithm
of	O
the	O
two-scale	O
equation	O
,	O
which	O
characterizes	O
refinable	O
functions	O
.	O
</s>
<s>
Refinable	O
functions	O
play	O
an	O
important	O
role	O
in	O
wavelet	O
theory	O
and	O
finite	B-Application
element	I-Application
theory	O
.	O
</s>
<s>
The	O
effect	O
of	O
can	O
be	O
expressed	O
in	O
terms	O
of	O
the	O
downsampling	B-Algorithm
operator	O
""	O
:	O
</s>
<s>
If	O
you	O
drop	O
the	O
first	O
and	O
the	O
last	O
column	O
and	O
move	O
the	O
odd-indexed	O
columns	O
to	O
the	O
left	O
and	O
the	O
even-indexed	O
columns	O
to	O
the	O
right	O
,	O
then	O
you	O
obtain	O
a	O
transposed	O
Sylvester	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
The	O
determinant	O
of	O
a	O
transfer	B-Algorithm
matrix	I-Algorithm
is	O
essentially	O
a	O
resultant	O
.	O
</s>
<s>
This	O
property	O
also	O
tells	O
,	O
how	O
vectors	O
from	O
the	O
null	B-Algorithm
space	I-Algorithm
of	O
can	O
be	O
converted	O
to	O
null	B-Algorithm
space	I-Algorithm
vectors	O
of	O
.	O
</s>
<s>
Actually	O
not	O
convolutions	B-Language
are	O
necessary	O
,	O
but	O
only	O
ones	O
,	O
when	O
applying	O
the	O
strategy	O
of	O
efficient	O
computation	O
of	O
powers	O
.	O
</s>
