<s>
In	O
mathematics	O
,	O
the	O
Zak	B-Algorithm
transform	I-Algorithm
(	O
also	O
known	O
as	O
the	O
Gelfand	O
mapping	O
)	O
is	O
a	O
certain	O
operation	O
which	O
takes	O
as	O
input	O
a	O
function	O
of	O
one	O
variable	O
and	O
produces	O
as	O
output	O
a	O
function	O
of	O
two	O
variables	O
.	O
</s>
<s>
The	O
output	O
function	O
is	O
called	O
the	O
Zak	B-Algorithm
transform	I-Algorithm
of	O
the	O
input	O
function	O
.	O
</s>
<s>
The	O
transform	O
is	O
defined	O
as	O
an	O
infinite	O
series	O
in	O
which	O
each	O
term	O
is	O
a	O
product	O
of	O
a	O
dilation	B-Algorithm
of	O
a	O
translation	B-Algorithm
by	O
an	O
integer	O
of	O
the	O
function	O
and	O
an	O
exponential	O
function	O
.	O
</s>
<s>
In	O
applications	O
of	O
Zak	B-Algorithm
transform	I-Algorithm
to	O
signal	O
processing	O
the	O
input	O
function	O
represents	O
a	O
signal	O
and	O
the	O
transform	O
will	O
be	O
a	O
mixed	O
time	O
–	O
frequency	O
representation	O
of	O
the	O
signal	O
.	O
</s>
<s>
The	O
Zak	B-Algorithm
transform	I-Algorithm
is	O
a	O
generalization	O
of	O
the	O
discrete	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
The	O
Zak	B-Algorithm
transform	I-Algorithm
had	O
been	O
discovered	O
by	O
several	O
people	O
in	O
different	O
fields	O
and	O
was	O
called	O
by	O
different	O
names	O
.	O
</s>
<s>
It	O
was	O
called	O
the	O
"	O
Gelfand	O
mapping	O
"	O
because	O
Israel	O
Gelfand	O
introduced	O
it	O
in	O
his	O
work	O
on	O
eigenfunction	B-Algorithm
expansions	I-Algorithm
.	O
</s>
<s>
There	O
seems	O
to	O
be	O
a	O
general	O
consensus	O
among	O
experts	O
in	O
the	O
field	O
to	O
call	O
it	O
the	O
Zak	B-Algorithm
transform	I-Algorithm
,	O
since	O
Zak	O
was	O
the	O
first	O
to	O
systematically	O
study	O
that	O
transform	O
in	O
a	O
more	O
general	O
setting	O
and	O
recognize	O
its	O
usefulness	O
.	O
</s>
<s>
In	O
defining	O
the	O
continuous-time	O
Zak	B-Algorithm
transform	I-Algorithm
,	O
the	O
input	O
function	O
is	O
a	O
function	O
of	O
a	O
real	O
variable	O
.	O
</s>
<s>
So	O
,	O
let	O
f(t )	O
be	O
a	O
function	O
of	O
a	O
real	O
variable	O
t	O
.	O
The	O
continuous-time	O
Zak	B-Algorithm
transform	I-Algorithm
of	O
f(t )	O
is	O
a	O
function	O
of	O
two	O
real	O
variables	O
one	O
of	O
which	O
is	O
t	O
.	O
The	O
other	O
variable	O
may	O
be	O
denoted	O
by	O
w	O
.	O
The	O
continuous-time	O
Zak	B-Algorithm
transform	I-Algorithm
has	O
been	O
defined	O
variously	O
.	O
</s>
<s>
The	O
special	O
case	O
of	O
Definition	O
1	O
obtained	O
by	O
taking	O
a	O
=	O
1	O
is	O
sometimes	O
taken	O
as	O
the	O
definition	O
of	O
the	O
Zak	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
In	O
this	O
special	O
case	O
,	O
the	O
Zak	B-Algorithm
transform	I-Algorithm
of	O
f(t )	O
is	O
denoted	O
by	O
Z[f]	O
.	O
</s>
<s>
The	O
notation	O
Z[f]	O
is	O
used	O
to	O
denote	O
another	O
form	O
of	O
the	O
Zak	B-Algorithm
transform	I-Algorithm
.	O
</s>
<s>
In	O
this	O
form	O
,	O
the	O
Zak	B-Algorithm
transform	I-Algorithm
of	O
f(t )	O
is	O
defined	O
as	O
follows	O
:	O
</s>
<s>
In	O
the	O
following	O
it	O
will	O
be	O
assumed	O
that	O
the	O
Zak	B-Algorithm
transform	I-Algorithm
is	O
as	O
given	O
in	O
Definition	O
2	O
.	O
</s>
<s>
Let	O
denote	O
convolution	B-Language
with	O
respect	O
to	O
the	O
variable	O
t	O
.	O
</s>
<s>
Given	O
the	O
Zak	B-Algorithm
transform	I-Algorithm
of	O
a	O
function	O
,	O
the	O
function	O
can	O
be	O
reconstructed	O
using	O
the	O
following	O
formula	O
:	O
</s>
<s>
The	O
discrete	O
Zak	B-Algorithm
transform	I-Algorithm
of	O
is	O
a	O
function	O
of	O
two	O
real	O
variables	O
,	O
one	O
of	O
which	O
is	O
the	O
integer	O
variable	O
.	O
</s>
<s>
The	O
discrete	O
Zak	B-Algorithm
transform	I-Algorithm
has	O
also	O
been	O
defined	O
variously	O
.	O
</s>
<s>
The	O
Zak	B-Algorithm
transform	I-Algorithm
has	O
been	O
successfully	O
used	O
in	O
physics	O
in	O
quantum	O
field	O
theory	O
,	O
in	O
electrical	O
engineering	O
in	O
time-frequency	O
representation	O
of	O
signals	O
,	O
and	O
in	O
digital	O
data	O
transmission	O
.	O
</s>
<s>
The	O
Zak	B-Algorithm
transform	I-Algorithm
has	O
also	O
applications	O
in	O
mathematics	O
.	O
</s>
