Bernstein polynomials are a classical tool in Computer Aided Design to create smooth maps with a high degree of local control. They are used for the construction of Bézier surfaces, free-form deformations, and many other applications. However, classical Bernstein polynomials are only defined for simplices and parallelepipeds. These can in general not directly capture the shape of arbitrary obje...

In this article, we will discuss various issues concerning when a complete Riemannian manifold possesses a global mean value inequality for positive subsolutions of either the Laplace equation or the heat equation. This study is motivated by the recent result of the first author [L1]. In that paper, he proved estimates on the dimensions of spaces of harmonic functions of at most polynomial grow...

We derive a generalization of barycentric coordinates which allows a vertex in a planar triangulation to be expressed as a convex combination of its neighbouring vertices. The coordinates are motivated by the Mean Value Theorem for harmonic functions and can be used to simplify and improve methods for parameterization and morphing.  2003 Elsevier Science B.V. All rights reserved.

Transfinite mean value interpolation has recently emerged as a simple and robust way to interpolate a function f defined on the boundary of a planar domain. In this paper we study basic properties of the interpolant, including sufficient conditions on the boundary of the domain to guarantee interpolation when f is continuous. Then, by deriving the normal derivative of the interpolant and of a m...