<s>
In	O
the	O
mathematical	O
field	O
of	O
numerical	B-General_Concept
analysis	I-General_Concept
,	O
discrete	B-Algorithm
spline	I-Algorithm
interpolation	I-Algorithm
is	O
a	O
form	O
of	O
interpolation	B-Algorithm
where	O
the	O
interpolant	O
is	O
a	O
special	O
type	O
of	O
piecewise	B-Algorithm
polynomial	I-Algorithm
called	O
a	O
discrete	B-Algorithm
spline	I-Algorithm
.	O
</s>
<s>
A	O
discrete	B-Algorithm
spline	I-Algorithm
is	O
a	O
piecewise	B-Algorithm
polynomial	I-Algorithm
such	O
that	O
its	O
central	B-Algorithm
differences	I-Algorithm
are	O
continuous	O
at	O
the	O
knots	B-Algorithm
whereas	O
a	O
spline	B-Algorithm
is	O
a	O
piecewise	B-Algorithm
polynomial	I-Algorithm
such	O
that	O
its	O
derivatives	B-Algorithm
are	O
continuous	O
at	O
the	O
knots	B-Algorithm
.	O
</s>
<s>
Discrete	B-Algorithm
cubic	I-Algorithm
splines	I-Algorithm
are	O
discrete	B-Algorithm
splines	I-Algorithm
where	O
the	O
central	B-Algorithm
differences	I-Algorithm
of	O
orders	O
0	O
,	O
1	O
,	O
and	O
2	O
are	O
required	O
to	O
be	O
continuous	O
.	O
</s>
<s>
Discrete	B-Algorithm
splines	I-Algorithm
were	O
introduced	O
by	O
Mangasarin	O
and	O
Schumaker	O
in	O
1971	O
as	O
solutions	O
of	O
certain	O
minimization	O
problems	O
involving	O
differences	O
.	O
</s>
<s>
then	O
g(x )	O
is	O
called	O
a	O
discrete	B-Algorithm
cubic	I-Algorithm
spline	I-Algorithm
.	O
</s>
<s>
The	O
conditions	O
defining	O
a	O
discrete	B-Algorithm
cubic	I-Algorithm
spline	I-Algorithm
are	O
equivalent	O
to	O
the	O
following	O
:	O
</s>
<s>
The	O
central	B-Algorithm
differences	I-Algorithm
of	O
orders	O
0	O
,	O
1	O
,	O
and	O
2	O
of	O
a	O
function	O
f(x )	O
are	O
defined	O
as	O
follows	O
:	O
</s>
<s>
This	O
states	O
that	O
the	O
central	B-Algorithm
differences	I-Algorithm
are	O
continuous	O
at	O
xi	O
.	O
</s>
<s>
The	O
following	O
function	O
defines	O
a	O
discrete	B-Algorithm
cubic	I-Algorithm
spline	I-Algorithm
:	O
</s>
<s>
Then	O
there	O
is	O
a	O
unique	O
cubic	O
discrete	B-Algorithm
spline	I-Algorithm
g(x )	O
satisfying	O
the	O
following	O
conditions	O
:	O
</s>
<s>
This	O
unique	O
discrete	B-Algorithm
cubic	I-Algorithm
spline	I-Algorithm
is	O
the	O
discrete	B-Algorithm
spline	I-Algorithm
interpolant	O
to	O
f(x )	O
in	O
the	O
interval	O
[	O
x0	O
-	O
h	O
,	O
xn	O
+	O
h ]	O
.	O
</s>
<s>
Discrete	B-Algorithm
cubic	I-Algorithm
splines	I-Algorithm
were	O
originally	O
introduced	O
as	O
solutions	O
of	O
certain	O
minimization	O
problems	O
.	O
</s>
<s>
They	O
have	O
applications	O
in	O
computing	O
nonlinear	O
splines	B-Algorithm
.	O
</s>
<s>
Discrete	O
interpolatory	O
splines	B-Algorithm
have	O
been	O
used	O
to	O
construct	O
biorthogonal	O
wavelets	O
.	O
</s>
