B-Spline Properties
Since
is nonzero only on the interval
, the linear
system for the B-spline coefficients of the spline to be determined,
by interpolation or least squares approximation, or even as the approximate
solution of some differential equation, is banded,
making the solving of that linear system particularly easy. For example,
if a spline s of order
with knot sequence
is
to be constructed so that
for
, then we are led to the linear
system
for the unknown B-spline coefficients
in which each
equation has at most
nonzero entries.
Also, many theoretical facts concerning splines are most easily
stated and/or proved in terms of B-splines. For example, it is possible
to match arbitrary data at sites
uniquely by
a spline of order
with knot sequence
if and only
if
for all
(Schoenberg-Whitney Conditions).
Computations with B-splines are facilitated by stable recurrence relations
that are also of help in the conversion from
B-form to ppform. The dual functional
provides a useful expression for the jth
B-spline coefficient of the spline s in terms
of its value and derivatives at an arbitrary site τ between
and
, and with
. It can be used
to show that
is closely related to
on the interval
, and seems the
most efficient means for converting from ppform to B-form.
 | Knot Multiplicity | | Constructive vs. Variational |  |
Includes the most popular MATLAB recorded presentations with Q&A sessions led by MATLAB experts.
Get the Interactive Kit
Store 
