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

• Published in: Applied mathematics and computation Vol. 500; p. 129417
• Main Authors: Bejancu, Aurelian, Dekhil, Mohamed
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.09.2025
• Subjects: Approximation order; End conditions; L-spline; Tension spline; Approximation order; Tension spline; 41A15; 41A05; 65D07; L-spline; End conditions
• ISSN: 0096-3003
• DOI: 10.1016/j.amc.2025.129417

For 0≤α≤β, let L=(D2−α2)(D2−β2), the Euler operator of the quadratic functional∫R{|D2f(t)|2+(α2+β2)|Df(t)|2+α2β2|f(t)|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<α=β, corresponding to Sobolev spline (or Matérn kernel) interpolation. For 0=α<β, which is the special case of tension splines, our natural L-spline interpolant with adjoint end conditions can be identified as an “Lm,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. •Unified setting for interpolation with a class of 4th order exponential L-splines.•Tension splines and Sobolev splines are prominent members of the L-spline class.•The L-spline interpolant minimizes an energy integral over the full real line.•The natural end conditions are expressed in terms of adjoint factors of L.•Improving on previous convergence results for full-line tension splines.