Boolean sum smooth interpolation to boundary data on a triangle is described. Sufficient conditions are given so that the functions when pieced together form a C N-1 (Ω) function over a triangular subdivision of a polygonal region Ω and the precision sets of the interpolation functions are derived. The interpolants are modified so that the compatability conditions on the function which is interpolated can be removed and a C 1 interpolant is used to illustrate the theory. The generation of interpolation schemes for discrete boundary data is also discussed.
Summary. Error bounds for interpolation remainders on triangles are derived by means of extensions of the Sard Kernel Theorems. These bounds are applied to the Galerkin method for elliptic boundary value problems. Certain kernels are shown to be identically zero under hypotheses which are, for example, fulfilled by tensor product interpolants on rectangles. This removes certain restrictions on how the sides of the triangles and/or rectangles tend to zero. Explicit error bounds are computed for piecewise linear interpolation over a triangulation and applied to a model problem.
Shepard's formula is an interpolation method for arbitrarily placed bivariate data. In this paper the continuity class and interpolation properties are proved. This is followed by generalizations which make the original method more useful. Graphical illustrations of the various methods conclude the paper.
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