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 extending to infinity: A. The real axis — oo <x < oo, y — 0. B. The half-plane — oo < # < oo, 0 < j < oo. The only poles of the approximating functions are to lie at preassigned points whose number will be required to be as small as possible. We shall make use of the theory of the class fgp the fundamental results of which are due to E. Hille and J. D. Tamarkin; &P is the set of functions F(z) which, for 0<y< oo, are regular and satisfy the inequality