Restricted uniform rational approximations

• Main Author: Borwein, Peter B.
• Language: English
• Published: 2010
• Online Access: http://hdl.handle.net/2429/21801

We consider the approximation of continuous functions from various restricted classes of rational functions. More precisely, for a function f continuous on [0,1] we consider [Mathematical Equation] where || || [0,1] is the uniform norm on [0,1] and A[sub n] is some restricted subset of the real rational functions of degree n . We are primarily concerned with the following three restrictions: monotone denominators, denominators having positive coefficients and both numerators and denominators having positive coefficients. We compare the order of approximation (the asymptotic behaviour of ε[sub n] ) from restricted rationals to the orders of approximation by polynomials and unrestricted rationals. We show, for functions that are exactly k-times differentiable on [0,1], that restricted rational approximation cannot be spectacularly more efficient than polynomial approximation. For example, if A[sub n] is the class of rational functions of degree n with monotone denominators and for some f the corresponding [Mathematical Equation] then f is k-times continuously differentiable on (0,1] . The situation is more complicated for analytic functions. In this case restricted rational approximations vary from being no more efficient than polynomials to being arbitrarily faster. For functions of the form ∑a[sub n]x[sup n], where a[sub n] ↓ 0, we show that the order of approximation by rationals with denominators having positive coefficients is the same as the polynomial order. Finally, we develop approximation theorems for rational functions with positive coefficients. (This class does not contain the polynomials so we cannot a priori assert denseness.) We show that it is possible to approximate positive functions fairly efficiently from this class. For instance, if f is analytic and positive on [0,1] then the order of approximation by rational functions of degree n with positive coefficients is at least as rapid as [Mathematical Equation], where ρ > 1 . === Science, Faculty of === Mathematics, Department of === Graduate