Lagrange's interpolation method is a simple and clever way of finding the unique
LLth-order polynomial that exactly passes through
L+1
L
1
distinct samples of a signal.
Once the polynomial is known, its value can easily be interpolated at any point
using the polynomial equation.
Lagrange interpolation is useful in many applications, including Parks-McClellan FIR Filter Design.

Note that for each term in the Lagrange interpolation formula above,
∏
i
=0,,i≠kLx−
x
i
x
k
−
x
i
={1 if x=
x
k
0 if (x=
x
j
)∧(j≠k)
i
0,
i
k
L
x
x
i
x
k
x
i
1
x
x
k
0
x
x
j
j
k
and that it is an L
Lth-order polynomial in x
x.
The Lagrange interpolation formula is thus exactly equal to
P
x
k
P
x
k
at all
x
k
x
k
,
and as a sum of LLth-order polynomials is itself an LLth-order polynomial.

It can be shown that the Vandermonde matrix
(
1
x
0
x
0
2...
x
0
L
1
x
1
x
1
2...
x
1
L
1
x
2
x
2
2...
x
2
L
⋮⋮⋮⋱⋮
1
x
L
x
L
2...
x
L
L
)
a
0
a
1
a
2
⋮
a
L
=P
x
0
P
x
1
P
x
2
⋮P
x
L
1
x
0
x
0
2
...
x
0
L
1
x
1
x
1
2
...
x
1
L
1
x
2
x
2
2
...
x
2
L
⋮
⋮
⋮
⋱
⋮
1
x
L
x
L
2
...
x
L
L
a
0
a
1
a
2
⋮
a
L
P
x
0
P
x
1
P
x
2
⋮
P
x
L
has a non-zero determinant and is thus invertible, so the
L
Lth-order polynomial passing through all
L+1
L
1
sample points
x
j
x
j
is unique. Thus the Lagrange polynomial expressions, as an L
Lth-order polynomial passing through the
L+1
L
1
sample points, must be the unique
Px
P
x
.