<s>
In	O
applied	O
mathematics	O
,	O
the	O
discrete	B-Algorithm
Chebyshev	I-Algorithm
transform	I-Algorithm
(	O
DCT	B-General_Concept
)	O
,	O
named	O
after	O
Pafnuty	O
Chebyshev	O
,	O
is	O
either	O
of	O
two	O
main	O
varieties	O
of	O
DCTs	B-General_Concept
:	O
the	O
discrete	B-Algorithm
Chebyshev	I-Algorithm
transform	I-Algorithm
on	O
the	O
'	O
roots	O
 '	O
grid	O
of	O
the	O
Chebyshev	O
polynomials	O
of	O
the	O
first	O
kind	O
and	O
the	O
discrete	B-Algorithm
Chebyshev	I-Algorithm
transform	I-Algorithm
on	O
the	O
'	O
extrema	O
 '	O
grid	O
of	O
the	O
Chebyshev	O
polynomials	O
of	O
the	O
first	O
kind	O
.	O
</s>
<s>
The	O
discrete	B-Algorithm
chebyshev	I-Algorithm
transform	I-Algorithm
of	O
u(x )	O
at	O
the	O
points	O
is	O
given	O
by	O
:	O
</s>
<s>
This	O
can	O
readily	O
be	O
obtained	O
by	O
manipulating	O
the	O
input	O
arguments	O
to	O
a	O
discrete	B-General_Concept
cosine	I-General_Concept
transform	I-General_Concept
.	O
</s>
<s>
This	O
can	O
be	O
demonstrated	O
using	O
the	O
following	O
MATLAB	B-Language
code	O
:	O
</s>
<s>
The	O
discrete	B-General_Concept
cosine	I-General_Concept
transform	I-General_Concept
(	O
dct	B-General_Concept
)	O
is	O
in	O
fact	O
computed	O
using	O
a	O
fast	O
Fourier	O
transform	O
algorithm	O
in	O
MATLAB	B-Language
.	O
</s>
<s>
And	O
the	O
inverse	O
transform	O
is	O
given	O
by	O
the	O
MATLAB	B-Language
code	O
:	O
</s>
<s>
There	O
is	O
a	O
discrete	O
(	O
and	O
in	O
fact	O
fast	O
because	O
it	O
performs	O
the	O
dct	B-General_Concept
by	O
using	O
a	O
fast	O
Fourier	O
transform	O
)	O
algorithm	O
available	O
at	O
the	O
MATLAB	B-Language
file	O
exchange	O
that	O
was	O
created	O
by	O
Greg	O
von	O
Winckel	O
.	O
</s>
<s>
The	O
primary	O
uses	O
of	O
the	O
discrete	B-Algorithm
Chebyshev	I-Algorithm
transform	I-Algorithm
are	O
numerical	O
integration	O
,	O
interpolation	O
,	O
and	O
stable	O
numerical	O
differentiation	O
.	O
</s>
