Polynomials are the approximating functions of choice when a smooth function is to be approximated locally. For example, the truncated Taylor series

$$\sum _{i=0}^{n}{\left(x-a\right)}^{i}{D}^{i}f(a)/i!$$

provides a satisfactory approximation for *f(x)* if *f* is
sufficiently smooth and *x* is sufficiently close
to *a*. But if a function is to be approximated on
a larger interval, the degree, *n*, of the approximating
polynomial may have to be chosen unacceptably large. The alternative
is to subdivide the interval *[a..b]* of approximation
into sufficiently small intervals [ξ_{j}..ξ_{j+1}],
with *a* = ξ_{1}<···
<ξ_{l+1} = *b*,
so that, on each such interval, a polynomial *p _{j}* of
relatively low degree can provide a good approximation to

There are two commonly used ways to represent a polynomial spline,
the ppform and the B-form. In this toolbox, a spline in ppform
is often referred to as a *piecewise polynomial*,
while a piecewise polynomial in B-form is often referred to as a spline.
This reflects the fact that piecewise polynomials and (polynomial)
splines are just two different views of the same thing.

The *ppform* of a polynomial spline of *order* *k* provides
a description in terms of its *breaks* ξ_{1}..ξ_{l+1} and
the *local polynomial
coefficients* *c _{ji}* of
its

$$\begin{array}{cc}{p}_{j}\left(x\right)={{\displaystyle \sum _{i=1}^{k}\left(x-{\xi}_{j}\right)}}^{k-i}{c}_{ji},& j=1:l\end{array}$$

For example, a cubic spline is of order 4, corresponding to
the fact that it requires four coefficients to specify a cubic polynomial.
The ppform is convenient for the evaluation and other *uses* of
a spline.

The *B-form* has become the standard way
to represent a spline during its *construction*,
because the B-form makes it easy to build in smoothness requirements across breaks and leads to banded
linear systems. The B-form describes a spline as a weighted sum

$$\sum _{j=1}^{n}{B}_{j,k}{a}_{j}$$

of B-splines of the required order *k*, with
their number, *n*, at least as big as *k*–1
plus the number of polynomial pieces that make up the spline. Here, *B*_{j,k }=* B* (·|*t _{j}*,
...,

$$\begin{array}{ccc}{\displaystyle \sum _{j=1}^{n}{B}_{j,k}\left(x\right)}=1& on& \left[{t}_{k}\mathrm{..}{t}_{n+1}\right]\end{array}$$

The multiplicity of the knots governs the smoothness, in the following way: If the
number τ occurs exactly *r* times in the sequence t_{j},...t_{j+k},
then *B _{j,k}* and its first

`bspligui`

.