The Carathéodory–Fejér Method for Real Rational Approximation

A “Carathéodory–Fejér method” is presented for near-best real rational approximation on intervals, based on the eigenvalue (or singular value) analysis of a Hankel matrix of Chebyshev coefficients. In approximation of a smooth function F, the CF approximant Rᶜᶠ frequently differs from the best approximation R^ * by only one part in millions or billions. To account for this we show here under weak assumptions that if F is approximated on [ - ε ,ε ], then as ε → 0, ||F - R^ * || = O(ε m + n + 1 ) while ||Rᶜᶠ - R^ * || = O(ε 3m + 2n + 3 ). In contrast, the latter figure would be O(ε m + n + 2 ) for the Chebyshev economization approximant of Maehly or the Chebyshev–Padé approximant of Gragg. It follows that as ε → 0, best approximation error curves approach the real parts of $m + n + 1$-winding rational functions of constant modulus to within O(ε 3m + 2n + 3 ). Numerical examples are given, including applications to eˣ on $[ - 1,1]$ and e- x on [0,∞ ). For the latter problem we conjecture that the errors in $(n,n)$ approximation decrease with each n by a ratio approaching a fixed constant 9.28903 ⋯.