Univariate interpolation for a class of L-splines with adjoint natural end conditions

For $0\le \alpha \le \beta$, let $L=(D^2-{\alpha }^2)(D^2-{\beta }^2)$, the Euler operator of the quadratic functional$\underset{R}{\int }\left\{\right|D^{2}f\left(t\right){|}^2+({\alpha }^2+{\beta }^2)\left|Df\right(t){|}^2+{\alpha }^2{\beta }^2|f\left(t\right){|}^2\}dt,$ where D is the first derivative operator. Given arbitrary values to be interpolated at a finite knot-set, we prove the existence of a unique L-spline interpolant from the natural space of functions f, for which the functional is finite. The natural L-spline interpolant satisfies adjoint differential conditions outside and at the end points of the interval spanned by the knot-set, and it is in fact the unique minimizer of the functional, subject to the interpolation conditions. This extends the approach by Bejancu (2011) for $0<\alpha =\beta$, corresponding to Sobolev spline (or Matérn kernel) interpolation. For $0=\alpha <\beta$, which is the special case of tension splines, our natural L-spline interpolant with adjoint end conditions can be identified as an “$L^{m,l,s}$-spline interpolant in $R$” (for $m=l=1$, $s=0$), previously studied by Le Méhauté and Bouhamidi (1992) via reproducing kernel theory. Our L-spline error analysis, confirmed by numerical tests, is improving on previous convergence results for such tension splines.