Function approximation

Let's start by defining the framework. Let's recall the definition: a function is a rule associating with any element of a definition set an element of an image set. In general, the definition set is a continuous subset of the real numbers, i.e. infinite and uncountable. The amount of information encompassed by the concept of a function is therefore beyond comprehension. (The elementary functions we can work with on paper make up only a tiny fraction of the whole set of functions, the few exceptions confirming the rule). In practical analytical problems, such as solving a differential equation, the unknown to be determined is or depends on a function. The information sought is therefore uncountable and the task impossible. Of course, the more differentiability is required, the more the number of possibilities decreases: for example, if we know from theory that the solution is integral, and if we can determine the countable set of its derivatives at a point, then it is known everywhere thanks to its Taylor series; the determination of the information is thus reduced to the development of an algorithm comprising a countable number of operations. Nevertheless, information remains infinite. The main task of approximation is to replace the infinite information contained in the functions with finite information determined by the fewest possible degrees of freedom.

In this article, we will be dealing with functions defined on an interval, finite or infinite; the problem of approximating functions of several variables will not be addressed. What's more, we're interested in approximation from a practical point of view, not in the theory of approximation. On the one hand, such functions can be horrible (e.g., derivable at any point), which is hardly the case in practice, and on the other hand, convergence is slow, not taking advantage of the reduction in possibilities induced by the existence of derivatives. Readers will find a wealth of theoretical literature, such as for a recent publication.

We will also restrict ourselves to infinitely derivable approximations, which we believe have many advantages. On the one hand, many of the functions used in practice are infinitely derivable, except perhaps at the extremities of their interval of definition; on the other hand, if the function has only a finite number of derivatives, the speed of convergence of the infinitely derivable interpolant automatically adapts to the highest order of differentiability; finally, the introduction of abscissas where the approximation has fewer derivatives than the approximated function seems to...