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

Aurelian Bejancu and Mohamed Dekhil

Applied Mathematics and Computation, 2025, vol. 500, issue C

Abstract: 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.

Keywords: Tension spline; L-spline; End conditions; Approximation order (search for similar items in EconPapers)
Date: 2025
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Persistent link: https://EconPapers.repec.org/RePEc:eee:apmaco:v:500:y:2025:i:c:s0096300325001444

DOI: 10.1016/j.amc.2025.129417

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