Approximation of unbounded functions via compactification

Approximants to functions f(s) that are allowed to possess infinite limits on their interval of definition, are constructed.To this end a compactification of Rⁿ is developed which is based on the projection of Rⁿ on a bowl-shaped subset of a parabolic surface. This compactification induces a bijection and a metric with several desirable properties that make it a useful tool for rational approximation of unbounded functions.Roughly speaking this compactification enables us to show that unbounded functions can be approximated by rational functions on a closed interval; thus we also obtain an extension to Weierstrass’ celebrated theorem. An extension to a Fourier-type theorem is also obtained. Roughly speaking, our result states that unbounded periodic functions can be approximated by quotients of certain trigonometric sums. The characteristics of the main results are the following. The approximations do not require the original approximated function to possess a restricted rate of growth. Neither do they require that the approximated function possess any amount of smoothness. Moreover, the numerator and denominator, in an approximating quotient are guaranteed not to vanish simultaneously. Furthermore, some of the proposed approximations are guaranteed to be bounded at every point at which the original approximated function is bounded. Beside the tool of compactification we also employ Bernstein polynomials and Cesaro means of “trigonometric sums”.