  Duchon's method of thin plate splines defines a polyharmonic interpolant to
scattered data values as the minimizer of a certain integral functional. For
transfinite interpolation, i.e. interpolation of continuous data prescribed on
curves or hypersurfaces, Kounchev has developed the method of polysplines,
which are piecewise polyharmonic functions of fixed smoothness across the given
hypersurfaces and satisfy some boundary conditions. Recently, Bejancu has
introduced boundary conditions of Beppo Levi type to construct a semi-cardinal
model for polyspline interpolation to data on an infinite set of parallel
hyperplanes. The present paper proves that, for periodic data on a finite set
of parallel hyperplanes, the polyspline interpolant satisfying Beppo Levi
boundary conditions is in fact a thin plate spline, i.e. it minimizes a Duchon
type functional.
