Abstract
This chapter discusses data and smoothness concepts related to two dimensional cases as per polyharmonic paradigm. The data concept of the polyharmonic paradigm states that the data set and the set of singularities coincide—that is, just as in the one-dimensional case of odd-degree splines, data set = knot set = singularity set. This means that the splines that will be constructed will consist of analytic pieces between the surfaces Γj and the interpolation and the smoothness conditions will be imposed solely on ∪ Γj. This is close to the one-dimensional situation of odd-degree splines where the data set and the set of the knot-points coincide, excluding the boundary that only carries data. This chapter also elaborated data concepts in two dimensions according to the polyharmonic paradigm, “parallel lines” or “strips,” and “concentric circles” or “annuli.” The symbolic map showing how the one-dimensional case is generalized to the two-dimensional case is illustrated. Smoothness concept according to the polyharmonic paradigm is also elaborated in this chapter.
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