Approximating a function by choosing a covering of its domain and ? points from its range
Suppose that we would like to compute the value of a real-valued function cp for every point 8 that might be selected from some given set E, but that this is infeasible and that we have to settle for approximation in the following manner: a finite covering’ of E, denoted C and containing j sets-to be called cells-is chosen. To each cell of C we assign a real number, called an outcome, from a k-element set A C R of possible outcomes. Given the 6 that is currently of interest we find a cell containing that 8, and we take as our estimate of q(B) the outcome assigned to that cell. We define the error of this procedure as the supremum, over all 8 in E, of the possible distances between the outcome so obtained and the true value q(B). Our objective is to minimize error by a suitable choice of C, of A, and of the assignment rule. Now given C and A, the choice of assignment rule is a simple matter, It