The univariate polynomial B-spline is a real-valued function on
a closed interval 
defined with the aid of a sequence of non decreasing
points, the so-called knot sequence 
with 
.
B-splines are defined using divided differences. Given an integer 
and a non-decreasing knot sequence 
, the 
th B-spline
of order k for the knot sequence 
is denoted by 
and is given
by [14]:
where 
is the piecewise truncated power function
of order k [14]:
The set of B-splines forms a basis that spans a vectorial
space 
. Any polynomial spline function 
of order k in
can be formulated as a linear combination of the B-splines
basis functions :
The coefficients 
can be selected so as to enforce specific
conditions on the spline function. For instance by enforcing that the spline
function agrees with a given data set values
(
) we get the spline interpolation
conditions:
This is a linear system of equations in the coefficients 
's.
The system is invertible provided that:
Similar linear system can be setup for other conditions such as least squares approximation.
The imposed conditions can be generalized through the use of some bounded
nonlinear functional 
and the nonlinear least squares objective as
follows:
where the 
's are the observed functional values and
a properly selected subspace with basis function
. Choosing an appropriate subspace is crucial.
The degree 
and general qualitative properties such as the smoothness
of the function 
can be determined based on expectations
about the solution in a straightforward fashion. However,
specifying the appropriate knot sequence () is a more intricate
and difficult problem. No algorithmic solution to the knot placement
problem exists in general for the nonlinear problem. Problem specific
strategies can be formulated based on heuristics and prior knowledge
of the physical laws involved (e.g. [57]).
Once the proper subspace is specified the solution of
(2.76) reduces to the determination of appropriate
coefficients that will minimize the least squares objective. This is a
discrete parameters extraction problem which can be solved by standard
nonlinear optimization techniques.
Spline models of the form of (2.73) are non-parameteric
models. A non-parameteric regression model assumes general
qualitative properties of the underlying function which are satisfied
if it is selected from a collection of functions such as
above. This allows greater flexibility in the possible form
of the function without the inherent assumptions necessary in
using parametric models. Nonparametric models rely more heavily
on data and are unbiased by a priori information.
Given a set of breakpoints 
for the spline function
, a recommended knot sequence is given for degree 
by:
This knot sequence ensures the continuity of and its 
first
derivatives in 
.
