Approximation and Interpolation of Functions

Often functions arising in economics and engineering are not specified by explicit formulas, but rather by values at distinct points. Engineering and mathematical analysis cannot easily be applied to problems having partially specified functions, and so interpolation functions of explicit and uncomplicated form are sought that agree with the known function values. In other instances, functions are completely specified, but have a form that cannot be transmitted conveniently to a computer. For in order to be describable to a computer, a function must be finitely representable. That is, it must be determined, with respect to whatever computer language is being used, by some finite set of numbers or instructions. The only real functions that are finitely representable in any of the computer languages are piecewise rational functions (ratios of polynomials). Ultimately then, if computer analysis involving other than a rational function is to be done, one must face the task of finding a rational function that provides a good approximation to the desired function.