Uniform approximation by rational functions having restricted denominators

AbstractThis paper considers approximation of continuous functions on a compact metric space by generalized rational functions for which the denominators have bounded coefficients and are bounded below by a fixed positive function. This lower bound alleviates numerical difficulties, and in some applications (e.g., digital filter design) has a useful physical interpretation. A “zero in the convex hull” characterization of best approximations is developed and used to prove uniqueness and de la Vallée Poussin results. Examples are given to illustrate this theory and its differences with the standard theory, where the denominators are merely required to be positive. A modified differential correction algorithm is presented and is proved to always converge at least linearly, and often quadratically