There are two grounds the spline theory stems from – the algebraic one (where splines are understood as piecewise smooth functions satisfying some continuity conditions) and the variational one (where splines are obtained via minimization of some quadratic functionals with constraints). We use the general variational approach called introduced by Talmi and Gilat and show that it covers not only the cubic spline and its 2D and 3D analogues but also the well known tension spline (called also spline with tension). We present the results of a 1D numerical example that characterize some properties of the tension spline.
@article{703005,
title = {A particular smooth interpolation that generates splines},
booktitle = {Programs and Algorithms of Numerical Mathematics},
series = {GDML\_Books},
publisher = {Institute of Mathematics CAS},
address = {Prague},
year = {2017},
pages = {112-119},
url = {http://dml.mathdoc.fr/item/703005}
}
Segeth, Karel. A particular smooth interpolation that generates splines, dans Programs and Algorithms of Numerical Mathematics, GDML_Books, (2017), pp. 112-119. http://gdmltest.u-ga.fr/item/703005/