An Algorithm for Real and Complex Rational Minimax Approximation

Real vs Complex Rational Chebyshev Approximation on an Interval

Real VS. Complex Rational Chebyshev Approximation on an Interval

I f f E C[-I, I] is real-valued, let Er( f ) and E'( f ) be the errors in best approximation to f in the supremum norm by rational functions of type ( m , n ) with real and complex coefficients, respectively. It has recently been observed that E'( f ) < Er( f ) can occur for any n > 1, but for no n 1 is it known whether y,,,, = inf, E'( f ) / E r ( f ) is zero or strictly positive. Here we show...

Rational Minimax Approximation via Adaptive Barycentric Representations

Computing rational minimax approximations can be very challenging when there are singularities on or near the interval of approximation — precisely the case where rational functions outperform polynomials by a landslide. We show that far more robust algorithms than previously available can be developed by making use of rational barycentric representations whose support points are chosen in an a...

Minimax principle and lower bounds in H2-rational approximation

We derive lower bounds in rational approximation of given degree to functions in the Hardy space H2 of the unit disk. We apply these to asymptotic error rates in rational approximation to Blaschke products and to Cauchy integrals on geodesic arcs. We also explain how to compute such bounds, either using Adamjan–Arov–Krein theory or linearized errors, and we present a couple of numerical experim...

A Unified Theory for Real vs Complex Rational Chebyshev Approximation on an Interval

A unified approach is presented for determining all the constants Ym.n (m > 0, n > 0) which occur in the study of real vs. complex rational Chebyshev approximation on an interval. In particular, it is shown that Ym,m+2 = 1/3 (m > 0), a problem which had remained open.