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I was reading some materials about spline funtions and there was something about knot duplication or knot repeating:

If we duplicate an interior knot, it transpires that the resulting basis still spans the space of piecewise polynomials but with one less continuous derivative at the knot. If we repeat that knot three times, then we have two fewer continuous derivatives there. For example, if the spline is cubic, repeating $e_j$ three times means that the first derivative is discontinuous at $e_j$; repeating it four times means that the spline itself is discontinuous at $e_j$. Thus for an Mth-order spline, repeating an interior knot M times means that there is a discontinuity at the knot.

This is from Linear Regression Analysis by Seber and Lee, page 175. I do not know what knot duplication or knot repeating means and how it affects the continuous derivatives. Could someone please shed some light on what it means or suggest me a reference in order to understand it?

Deepleeqe
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    See the excellent but difficult book of Carl de Boor A Practical Guide to Splines, chapter IX. Mutliple knots are well understood for an order 2 (broken line spline) where a double knot can be used to allow a step. When two successive distinct knots are very close, by drawing famous ``hat'' basis functions involving these knots we get a nearly vertical segment. Passing to the limit we get discontinuous functions with a specified location for step. – Yves Jul 05 '16 at 08:38
  • Knot duplication is where you place a second knot at the same x-value. – Glen_b Jul 05 '16 at 09:48
  • Thank you for your help!@Yves I tried to read the book you recommended but it is really difficult. I post a link which is a lecture about B spline https://www.youtube.com/watch?v=OkncKzflw8I – Deepleeqe Jul 05 '16 at 13:02
  • @Deepleeqe B-splines are a different thing than the Bezier spline, although a confusion of these two notions is quite common. You can experiment with the B-spline basis with given knots sequence using the splineBasis function of the splines R-package (recommended package usually coming with R), starting from the example in the help of this function. Taking ord = 2 or 3 and changing the knots sequence can be instructive. – Yves Jul 05 '16 at 14:25
  • Sorry the name of the function is splineDesign. – Yves Jul 05 '16 at 15:19

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