Chapter 6

Carl de Boor

This chapter promotes, details and exploits the fact that (univariate) splines, i.e., smooth piecewise polynomial functions, are weighted sums of B-splines.

A **piecewise polynomial** of **order** *k* with **break sequence ξ** (necessarily strictly increasing) is, by definition, any function *f* that, on each of the half-open intervals [ξ_{j} ξ_{j+1}), agrees with some polynomial of degree < *k*. The term ‘order’ used here is not standard but handy.

Note that this definition makes a piecewise polynomial function **right-continuous**, meaning that, for any *x, f*(*x*) = *f*(*x*+):= lim_{h↓0} *f*(*x* + *h*). This choice is ...

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