Functions of Closest Approximation on an Infinite Range
William DeW. Cairns · American Mathematical Monthly · 1927
The purpose of this paper is to extend to the interval (oo, oo) a theorem of Dunham Jackson2 on the existence of the polynomial of specified degree, or of a linear combination of n given functions, which gives the best approximation to a given continuous function in a given interval (a, b). This extension is accomplished by the expedient of introducing the factor e-Z'/4. The general argument is quite similar to that of the paper referred to; the analogy is shown more fully by the rather free use of this exponential factor. Let p,(x), p2(x), . , p^(x) be n functions, continuous for finite x, and such that one can find a positive constant h so that