A discrete variable representation (DVR) expresses a finite-dimensional vector space of functions in a vector basis associated with a set of nodes. It converts between expansion coefficients and weighted values at the nodes, allowing multiplication by a function to be approximated by a diagonal matrix (Light et al. 1985).
One construction uses an 
-point Gaussian quadrature
rule for a positive weighting function 
on an interval 
. Let 
be its nodes, 
its Christoffel numbers,
and 
,
,
..., 
the orthogonal polynomials with real coefficients, normalized in the weighted inner
product. The matrix with elements
 |
(1)
|
for 
,
2, ..., 
satisfies
 |
(2)
|
where 
is the identity matrix and 
denotes the transpose. This
follows because Gaussian quadrature integrates
each product 
, of polynomial degree
at most 
, exactly. Thus 
is an orthogonal matrix.
The transformed orthonormal basis consists of
 |
(3)
|
where 
are the fundamental Lagrange interpolating
polynomials for the nodes. They satisfy
 |
(4)
|
where 
is the Kronecker delta. Consequently, if 
,
its transformed expansion coefficients are
.
For multiplication by a function 
, the quadrature gives
 |
(5)
|
This diagonal matrix is exact whenever the integrands fall within the quadrature's exactness class. For a general 
, it is a quadrature approximation to the exact matrix
of multiplication on the trial subspace. The Lagrange-mesh
method uses this representation in approximating differential
operators.
