Best approximation by rational functions and by meromorphic functions with some free poles

Abstract

I have recently indicated [1, 2, 3, 4] some cases of best approximation of a meromorphic function f(z) by rational functions R.~(z) of a given type (n, v) having some free poles (that is, poles not prescribed in position), where it is proved that the free poles approach necessarily (n ~ ~) the poles of f (z) . The object of the present paper is to indicate (w that the methods already introduced for the case that the prescribed poles of the R.v(z) lie at infinity admit extensions that apply to the more general case that the prescribed poles of the R.v(z) do not lie at infinity nor in fact in a finite number of points. We study also (w the problem of approximation by meromorphic functions, bounded with the exception of v free poles.