In many applications we want to find an approximation for a function, for example for differential equations. Example problem: We want to understand how a calculator or computer can evaluate sinx for a given value x. The processor can essentially only perform addition, multiplication, division. Therefore we can try to approximate the function with a polynomial. We use the notation Pn for polyno...

We construct symmetric polar WAMs (Weakly Admissible Meshes) with low cardinality for least-squares polynomial approximation on the disk. These are then mapped to an arbitrary triangle. Numerical tests show that the growth of the least-squares projection uniform norm is much slower than the theoretical bound, and even slower than that of the Lebesgue constant of the best known interpolation poi...

Least squares approximation is a technique to find an approximate solution to a system of linear equations that has no exact solution. In a typical setting, one lets n be the number of constraints and d be the number of variables, with n d. Then, existing exact methods find a solution vector in O(nd2) time. We present two randomized algorithms that provide accurate relative-error approximations...

An alternative to radial basis function interpolation and approximation is the so-called moving least squares method. As we will see below, in this method the approximation Pf to f is obtained by solving many (small) linear systems, instead of via solution of a single – but large – linear system as we did in the previous chapters. To make a connection with the previous chapters we start with th...