Approximation by Rational Functions on Riemann Surfaces

Uniform Approximation on Riemann Surfaces

This thesis consists of three contributions to the theory of complex approximation on Riemann surfaces. It is known that if E is a closed subset of an open Riemann surface R and f is a holomorphic function on a neighbourhood of E, then it is “usually” not possible to approximate f uniformly by functions holomorphic on all of R. In Chapter 2, we show, however, that for every open Riemann surface...

Approximation by Rational Functions

Approximation by Rational Functions

Making use of the Hardy-Littlewood maximal function, we give a new proof of the following theorem of Pekarski: If f' is in L log L on a finite interval, then f can be approximated in the uniform norm by rational functions of degree n to an error 0(1/n) on that interval. It is well known that approximation by rational functions of degree n can produce a dramatically smaller error than that for p...

A Note on Approximation by Rational Functions

The theory of the approximation by rational functions on point sets E of the js-plane (z = x+iy) has been summarized by J. L. Walsh who himself has proved a great number of important theorems some of which are fundamental. The results concern both the case when E is bounded and when E extends to infinity. In the present note a Z^-theory (0<p< oo) will be given for the following point sets exten...

Approximation by Rational Functions in

The denseness of rational functions with prescribed poles in the Hardy space and disk algebra is considered. Notations. C complex plane D unit disk fz : jzj < 1g Tunit circle fz : jzj = 1g H p Hardy space of analytic functions on D kfk 1 := supfjf(z)j : z 2 D g, the H 1 norm A(D) disk algebra of functions analytic on D and continuous on D P n set of polynomials of degree at most n