What is $B_{i,n}(x)$ in the Wiki page for B-splines

Question

A spline is a piecewise polynomial function with k knots or places where the polynomial function can change between the knots. It has order n if its knots are denoted $t_0,t_1,...,t_n$. For a given knot sequence, there is a unique spline satisfying

$$B_{i,n}(x)=\begin{cases}0&if x<t_i orx\ge t_{i+n}\\ nonzero&otherwise\end{cases}$$

If we add the additional constraint that $\sum_i B_{i, n}(x)=1$ for all x between the first and last knot, then the scaling factor $B_{i,n}(x)$ becomes fixed. The resulting $B_{i,n}(x)$ functions are called B-splines.

The usefulness of B-splines lies in the fact that any spline function of order n on a given set of knots can be expressed as a linear combination of B-splines:

$$S_{n,\textbf t}(x)=\sum_{i}\alpha_iB_{i,n}(x)$$

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Can someone explain this to me? I understand very little of what's said.

• What exactly is $B_{i,n}(x)$ and what does the index i represent? Why is it zero between $t_i,t_{i+n}$? Does nonzero mean some curve, or is it a real number?

• Why do we sum the $\sum_{i}B_{i,n}$ to 1?

• I do understand that a set of basis functions is analogous to a basis for a vector space, in that there is a unique representation of a spline function given those basis functions. I guess the issue returns to what exactly is $B_{i,n}(x)$ and how can linear combinations of them sum up to any spline?

Answer 1

1. The $B_{i,n}$ form a basis for a given $n$ (the spline order). The splines are indexed by $i$. Typically, you would choose a $n$ and then build the basis over some grid, indexing all the elements of the basis by $i$.
2. The basis is not unique so it is normalized in order to have the sum of all elements equal to one. This is a convenience choice.
3. It's possible to prove that the basis can match any spline (of the same order). This is purely a convenience, you can still use normal splines and get the exact same results. However the B-splines are often simpler with computers.