More Scatt Red Data Smoothing

Abstract

Abstract A certain drawback of tensor product splines is that they are always defined on a rectangle. In applications with a well specified and non-rectangular approximation domain D their use may therefore be rather inconvenient. Of course, if D is bounded, we can always define a rectangle R containing D and then determine a spline approximation on R. However, this will not always give satisfactory results, especially if some additional constraints at the boundary of D must be fulfilled. Also, the inevitable fact that we will create (large) subregions of the rectangle without data points will increase the chance of rank deficient systems (see section 9,1.2) and may therefore endanger the numerical stability of the approximation algorithm. Another possibility is to search for a smooth transformation that maps the region D onto a rectangle R. Of course this will not always be possible. Besides, it may happen that we will have to impose additional constraints on the tensor product spline to guarantee that the corresponding function on D is still sufficiently smooth. As an illustration, we will show how the scattered data algorithims of Chapter 9 can be adapted for regions which can easily be described in polar coordinates. In particular, we will first consider the unit disk C and then, more generally, any region D that has a boundary which can be described by a smooth periodic function of the angular coordinate. As a second example we will consider the smoothing of data over the sphere. Here also additional boundary constraints will have to be imposed on the approximating tensor product splines, similar to those for the unit disk