We consider the piecewise linear approximation of saddle functions of the form (f(x,y)=ax^2-by^2 ) under the (L_ ) error norm. We show that interpolating approximations are not optimal. One can get slightly smaller errors by allowing the vertices of the approximation to move away from the graph of the function.
In arbitrary dimensions, the problem of optimal piecewise linear approximation has been adressed by Clarkson @cite_4 , without deriving explicit constant factors. For convex functions and convex bodies, there is a vast literature on optimal piecewise linear approximation in many variations, see for example the treatment in @cite_3 and the references given there.
@cite_4 @cite_3
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This work addresses the problem of approximating a manifold by a simplicial mesh, and the related problem of building triangulations for the purpose of piecewise-linear approximation of functions. It has long been understood that the vertices of such meshes or triangulations should be "well-distributed," or satisfy certain "sampling conditions." This work clarifies and extends some algorithms for finding such well-distributed vertices, by showing that they can be regarded as finding e-nets or Delone sets in appropriate metric spaces. In some cases where such Delone properties were already understood, such as for meshes to approximate smooth manifolds that bound convex bodies, the upper and lower bound results are extended to more general manifolds; in particular, under some general conditions, the minimum Hausdorff distance for a mesh with n simplices to a d-manifold M is Θ((∫M√|κ(x)| n)2 d) as n ⋺ ∞, where κ(x) is the Gaussian curvature at point x ∈ M. We also relate these constructions to Dudley's approximation scheme for convex bodies, which can be interpreted as involving an e-net in a metric space whose distance function depends on surface normals. We study piecewise linear approximation of quadratic functions de- flned on R n . Invariance properties and canonical Caley Klein metrics that help in understanding this problem can be handled in arbitrary dimensions. However, the problem of optimal approximants in the sense that their linear pieces are of maximal size by keeping a given error tolerance, is a di-cult one. We present a detailled discussion of the case n = 2, where we can partially use results from convex geometry and discrete geometry. The case n = 3 is considerably harder, and thus just a few results can be formulated so far.