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Construction of polygonal interpolants: a maximum entropy approach. (English) Zbl 1073.65505
Summary: We establish a link between maximizing (information-theoretic) entropy and the construction of polygonal interpolants. The determination of shape functions on $$n$$-gons $$(n>3)$$ leads to a non-unique under-determined system of linear equations. The barycentric co-ordinates $$\varphi_i$$, which form a partition of unity, are associated with discrete probability measures, and the linear reproducing conditions are the counterpart of the expectations of a linear function. The $$\varphi_i$$ are computed by maximizing the uncertainty $$H(\varphi_1,\varphi_2, \dots,\varphi_n)=-\sum^n_{i=1}\varphi_i\log\varphi_i$$, subject to the above constraints. The description is expository in nature, and the numerical results via the maximum entropy (MAXENT) formulation are compared to those obtained from a few distinct polygonal interpolants. The maximum entropy formulation leads to a feasible solution for $$\varphi_i$$ in any convex or non-convex polygon. This study is an instance of the application of the maximum entropy principle, wherein least-biased inference is made on the basis of incomplete information.

##### MSC:
 65D05 Numerical interpolation 41A05 Interpolation in approximation theory 94A17 Measures of information, entropy
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