When constructing interpolating polynomials, there is a tradeoff between having a better fit and having a smooth well-behaved fitting function. The more data points that are used in the interpolation, the higher the degree of the resulting polynomial, and therefore the greater oscillation it will exhibit between the data points. Therefore, a high-degree interpolation may be a poor predictor of the function between points, although the accuracy at the data points will be "perfect."

For
points,

(4)

(5)

Note that the function passes through the points , as can be seen for the case ,

(6)

(7)

(8)

Generalizing to arbitrary ,

(9)

The Lagrange interpolating polynomials can also be written using what Szegö (1975) called Lagrange's fundamental interpolating polynomials. Let

(10)

(11)

(12)

(13)

so that is an th degree polynomial with zeros
at ,
..., .
Then define the fundamental polynomials by

(14)

which satisfy

(15)

where
is the Kronecker delta. Now let , ..., , then the expansion

(16)

gives the unique Lagrange interpolating polynomial assuming the values at . More generally, let be an arbitrary distribution on the interval ,
the associated orthogonal polynomials,
and ,
...,
the fundamental polynomials corresponding to the set
of zeros of a polynomial . Then