Definition
(Lagrange interpolation formula)
For a set
we define
according to the
relationship


(Lagrange interpolation formula 1)

Note that
hence
Therefore
Indeed,
has the form
.
It is defined by values in
distinct points.
Proposition
(Coefficients of quadrature rule)
For a given set
there exist
such that the quadrature rule (
Quadrature rule
1
) is interpolatory.
Proposition
(Higher degree of exactness) Given an
integer
,
the formula (
Quadrature rule 1
) has degree
of exactness
iff both of the following conditions are satisfied:
1. The formula (
Quadrature rule 1
) is
interpolatory.
2. The
polynomial
satisfies
The coefficients
may be found from the proposition
(
Coefficients of quadrature
rule
).
