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

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.