Approximation by rational functions

Abstract

In this paper, we show that if Fe Lv(k, cx) on r' where r' denotes the border of a compact bordered Riemann surface R, then F can be uniquely written as the sum of a function in H(k, cx) and a function in Gv(k, cx) and moreover that F can be approximated on r' in norm to within A/nk+? by a sequence of rational functions on the union of P with its double. The purpose of this paper is to generalize some results of the second named author [4] concerning the Hardy classes HP and their particularizations to the classes H(k, c) of functions satisfying (together with certain derivatives) a Lipschitz condition to Riemann surfaces. Let y denote the unit circle. We define the class L(k, cx) (0< cc< 1, 1 <p< co) to be the set of functions possessing derivatives up to order k on y whose kth derivatives satisfy a pth mean integrated Lipschitz condition of order ac. H(k, ac) will denote the subclass of functions belonging to L(k, a) which belong to the Hardy class HP of the interior of y while G1(k, a) will denote those functions belonging to LP(k, cx) which are boundary functions of functions belonging to HP of the exterior of y and vanishing at so. In [4], it is shown that every function of the class LP(k, cx) can be written uniquely as the sum of a function in HP(k, oc) and a function in GP(k, cx). It is this result that we shall generalize to the setting of Riemann surfaces and in addition obtain some results on approximation by rational functions. Let R denote the interior of a compact bordered Riemann surface 1? with boundary r. Suppose f possesses derivatives up to order k on 1r. Let F1, * *, Fn denote the connected components of F and let Xi, i= 1, * , n, denote the uniformizers which map Fi onto y. We shall say thatf is of class Lp(k, cc) on F iffo xZ1 is of class L (k, cc) on y for all i. This definition is independent of the choice of uniformizing variables as a consequence of a theorem of Hardy and Littlewood [3]. Received by the editots February 4, 1972. AMS 1969 subject classifications. Primary 3070; Secondary 3067.