By John A. Hertz, Anders S. Krogh, Richard G. Palmer
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Extra info for Introduction To The Theory Of Neural Computation, Volume I (Santa Fe Institute Series)
Let x°, • • • ,x s G R , s > 1. The simplex with vertices x°, • • • ,x s is called an s-simplex, if its (signed) volume is nonzero. Here, x* = (zj, • • • , x l s ) . Suppose that ( x ° , - - - , x s ) is an g S'-simplex. Then any x = (zi, • • • , z s ) in R can be identified by an (s + l)-tuple (Ao, • • • , A s ), where This (s + l)-tuple is called the barycentric coordinate of x relative to the s-simplex (x°, • • • ,x s ). Note that each A^ — A^(x) is a linear polynomial in x.
In applications, however, we would like to use the smoothest splines with the lowest degree but, at the same time, be able to do the approximation. That is, we are interested in working with the spaces S£(AMN) wnere> for a given r e Z_|_, d is the smallest so that ard or brd are nonzero. We will use the notation d* — d*(r,i) for the smallest d such that ard > 0 for i = l and brd > 0 for i = 2 and denote by a*, 6* the corresponding values of a£, brd. 4), we have the following table. That is, for the three-directional mesh, there are one or two "independent" locally supported splines with minimal degree, while for the fourdirectional mesh, there are up to three "independent" locally supported ones with minimal degree, depending on the smoothness requirement.
Another interesting subspace is the space of periodic bivariate splines. In ter Morsche  the periodic spline subspace of SS(&MN) is studied and the property of unisolvence in interpolating periodic data is also discussed. The periodicity requirement leads naturally to approximation of Fourier coefficients of a periodic function by interpolating bivariate periodic splines. This topic is studied in ter Morsche , and we will delay our brief discussion of it to Chapter 9. CHAPTER 5 Bezier Representation and Smoothing Techniques In this chapter we will consider only grid partitions consisting of simplices and parallelepipeds.