A new class of interpolatory $L$-splines with adjoint end conditions
This work advances the theoretical foundation of polyspline methods for transfinite surface interpolation, relevant to researchers in approximation theory and geometric modeling.
The paper introduces a new class of interpolatory L-splines with adjoint end conditions, establishing existence, uniqueness, and variational properties for non-zero frequency cases. It provides an L²-error bound for transfinite surface interpolation using Beppo Levi polysplines.
A thin plate spline surface for interpolation of smooth transfinite data prescribed along concentric circles was recently proposed by Bejancu, using Kounchev's polyspline method. The construction of the new `Beppo Levi polyspline' surface reduces, via separation of variables, to that of a countable family of univariate $L$-splines, indexed by the frequency integer $k$. This paper establishes the existence, uniqueness and variational properties of the `Beppo Levi $L$-spline' schemes corresponding to non-zero frequencies $k$. In this case, the resulting $L$-spline end conditions are formulated in terms of \emph{adjoint} differential operators, unlike the usual `natural' $L$-spline end conditions, which employ identical operators at both ends. Our $L$-spline error analysis leads to an $L^{2}$-error bound for transfinite surface interpolation with Beppo Levi polysplines.