A new class of interpolatory $L$-splines with adjoint end conditions

TL;DR

This paper introduces a novel class of interpolatory $L$-splines with adjoint end conditions, providing theoretical foundations and error bounds for their use in transfinite surface interpolation.

Contribution

It establishes existence, uniqueness, and variational properties of Beppo Levi $L$-splines with adjoint end conditions, advancing spline theory for transfinite data.

Findings

Proves existence and uniqueness of the new $L$-splines.

Derives an $L^{2}$-error bound for transfinite surface interpolation.

Introduces adjoint differential operators in spline end conditions.

Abstract

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.