Non-strong uniqueness in real and complex Chebyshev 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...

Chebyshev Centers and Approximation in Pre-Hilbert C*-Modules

Near-Circularity of the Error Curve In Complex Chebyshev Approximation

Let f(z) be analytic on the unit disk, and let p*(z) be the best (Chebyshev) polynomial approximation to f(z) on the disk of degree at most n. It is observed that in typical problems the “error curve,” the image of the unit circle under (f p*)(z), often approximates to a startling degree a perfect circle with winding number n + 1. This phenomenon is approached by consideration of related proble...

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.