Real and complex Chebyshev approximation on the unit disk and interval

Rational Chebyshev Approximation on the Unit Disk

In a recent paper we showed that er ror curves in po lynomia l Chebyshev a p p r o x i m a t i o n of ana ly t ic functions on the unit disk tend to a p p r o x i m a t e perfect circles abou t the origin [23]. M a k i n g use of a theorem of Ca ra th6odo ry and Fej6r, we der ived in the process a me thod for calculat ing near-bes t a p p r o x i m a t i o n s rapid ly by finding the pr incipal...

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...

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.

Shape Preserving Approximation by Complex Polynomials in the Unit Disk

The purpose of this paper is to obtain new results concerning the preservation of some properties in Geometric Function Theory, in approximation of analytic functions by polynomials, with best approximation types of rates. In addition, the approximating polynomials satisfy some interpolation conditions too.