<s>
In	O
time	O
series	O
analysis	O
,	O
the	O
shift	B-Algorithm
operator	I-Algorithm
is	O
called	O
the	O
lag	O
operator	O
.	O
</s>
<s>
Shift	B-Algorithm
operators	I-Algorithm
are	O
examples	O
of	O
linear	B-Architecture
operators	I-Architecture
,	O
important	O
for	O
their	O
simplicity	O
and	O
natural	O
occurrence	O
.	O
</s>
<s>
The	O
shift	B-Algorithm
operator	I-Algorithm
action	O
on	O
functions	O
of	O
a	O
real	O
variable	O
plays	O
an	O
important	O
role	O
in	O
harmonic	O
analysis	O
,	O
for	O
example	O
,	O
it	O
appears	O
in	O
the	O
definitions	O
of	O
almost	O
periodic	O
functions	O
,	O
positive-definite	O
functions	O
,	O
derivatives	B-Algorithm
,	O
and	O
convolution	B-Language
.	O
</s>
<s>
Shifts	O
of	O
sequences	O
(	O
functions	O
of	O
an	O
integer	O
variable	O
)	O
appear	O
in	O
diverse	O
areas	O
such	O
as	O
Hardy	O
spaces	O
,	O
the	O
theory	O
of	O
abelian	O
varieties	O
,	O
and	O
the	O
theory	O
of	O
symbolic	O
dynamics	O
,	O
for	O
which	O
the	O
baker	B-Algorithm
's	I-Algorithm
map	I-Algorithm
is	O
an	O
explicit	O
representation	O
.	O
</s>
<s>
The	O
shift	B-Algorithm
operator	I-Algorithm
(	O
where	O
)	O
takes	O
a	O
function	O
on	O
R	O
to	O
its	O
translation	O
,	O
</s>
<s>
A	O
practical	O
operational	O
calculus	O
representation	O
of	O
the	O
linear	B-Architecture
operator	I-Architecture
in	O
terms	O
of	O
the	O
plain	O
derivative	B-Algorithm
was	O
introduced	O
by	O
Lagrange	O
,	O
</s>
<s>
The	O
right	O
and	O
left	O
shift	B-Algorithm
operators	I-Algorithm
acting	O
on	O
two-sided	O
infinite	O
sequences	O
are	O
called	O
bilateral	B-Algorithm
shifts	I-Algorithm
.	O
</s>
<s>
The	O
shift	B-Algorithm
operator	I-Algorithm
acting	O
on	O
real	O
-	O
or	O
complex-valued	O
functions	O
or	O
sequences	O
is	O
a	O
linear	B-Architecture
operator	I-Architecture
which	O
preserves	O
most	O
of	O
the	O
standard	O
norms	O
which	O
appear	O
in	O
functional	B-Application
analysis	I-Application
.	O
</s>
<s>
The	O
shift	B-Algorithm
operator	I-Algorithm
acting	O
on	O
two-sided	O
sequences	O
is	O
a	O
unitary	B-Algorithm
operator	I-Algorithm
on	O
.	O
</s>
<s>
The	O
shift	B-Algorithm
operator	I-Algorithm
acting	O
on	O
functions	O
of	O
a	O
real	O
variable	O
is	O
a	O
unitary	B-Algorithm
operator	I-Algorithm
on	O
.	O
</s>
<s>
In	O
both	O
cases	O
,	O
the	O
(	O
left	O
)	O
shift	B-Algorithm
operator	I-Algorithm
satisfies	O
the	O
following	O
commutation	O
relation	O
with	O
the	O
Fourier	O
transform	O
:	O
</s>
<s>
where	O
is	O
the	O
multiplication	B-Algorithm
operator	I-Algorithm
by	O
.	O
</s>
<s>
The	O
one-sided	O
shift	O
acting	O
on	O
is	O
a	O
proper	O
isometry	O
with	O
range	B-Algorithm
equal	O
to	O
all	O
vectors	O
which	O
vanish	O
in	O
the	O
first	O
coordinate	O
.	O
</s>
<s>
This	O
observation	O
is	O
at	O
the	O
heart	O
of	O
the	O
construction	O
of	O
many	O
unitary	B-Algorithm
dilations	I-Algorithm
of	O
isometries	O
.	O
</s>
<s>
Jean	O
Delsarte	O
introduced	O
the	O
notion	O
of	O
generalized	O
shift	B-Algorithm
operator	I-Algorithm
(	O
also	O
called	O
generalized	O
displacement	O
operator	O
)	O
;	O
it	O
was	O
further	O
developed	O
by	O
Boris	O
Levitan	O
.	O
</s>
<s>
A	O
family	O
of	O
operators	O
acting	O
on	O
a	O
space	O
of	O
functions	O
from	O
a	O
set	O
to	O
is	O
called	O
a	O
family	O
of	O
generalized	O
shift	B-Algorithm
operators	I-Algorithm
if	O
the	O
following	O
properties	O
hold	O
:	O
</s>
