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 Rcffrequently 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,$\parallel F-R^{\ast}\parallel =O(\varepsilon ^{m+n+1})$while$\parallel R^{cf}-R^{\ast}\parallel =O(\varepsilon ^{3m+2n+3})$. In contrast, the latter figure would be$O(\varepsilon ^{m+n+2})$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$O(\varepsilon ^{3m+2n+3})$. Numerical examples are given, including applications to$e^{x}$on [-1, 1] and e-xon [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 · · ·.