Existence of transformed rational complex Chebyshev approximations, II

Rational Chebyshev Approximations for Fermi-Dirac Integrals of Orders

Rational Chebyshev approximations are given for the complete Fermi-Dirac integrals of orders — \, \ and f. Maximal relative errors vary with the function and interval considered, but generally range down to 10~9 or less.

A Fast Algorithm for Linear Complex Chebyshev Approximations

We propose a new algorithm for finding best minimax polynomial approximations in the complex plane. The algorithm is the first satisfactory generalization of the well-known Remez algorithm for real approximations. Among all available algorithms, ours is the only quadratically convergent one. Numerical examples are presented to illustrate rapid convergence.

Computing rational Gauss-Chebyshev quadrature formulas with complex poles

We provide a fast algorithm to compute arbitrarily many nodes and weights for rational Gauss-Chebyshev quadrature formulas integrating exactly in spaces of rational functions with arbitrary complex poles outside [−1, 1]. This algorithm is based on the derivation of explicit expressions for the Chebyshev (para-)orthogonal rational functions.

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...