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## fulfill the (cubic) basis splines the partition of unity at all points i

**From**: |
Simon Wiesheier |

**Subject**: |
fulfill the (cubic) basis splines the partition of unity at all points in [a,b]? |

**Date**: |
Tue, 7 Mar 2023 17:21:42 +0100 |

Dear all,
my objective is to create cubic basis splines to approximate a function
f = c_i B_i (sum over i:=1,...,n)
over the interval [a,b].
The sum of the n basis functions B_i should be equal to one everywhere in
[a,b], that is, the basis functions should fulfill the partition of unity.
After reading the manual, it is not clear to me how GNU internally
constructs the knot vector.
There are the functions,
gsl_bspline_knots
gsl_bspline_knots_uniform,
that create the knot vector based on given breakpoints.
I would appreciate if someone gave me some hints how to create
the vector of breakpoints ,
const gsl_vector
<https://www.gnu.org/software/gsl/doc/html/vectors.html#c.gsl_vector> *
breakpts ,
in order to get my desired output.
I probably will have to expand [a,b] by some pseudo control-points c_i and,
when evaluating the spline f, restrict the interval again to [a,b].
Is this the right way to go?
Best
Simon

**fulfill the (cubic) basis splines the partition of unity at all points in [a,b]?**,
*Simon Wiesheier* **<=**