1. Acta mathemutica vol. 6 (1885) pp. 229–244.
2. For more detailed results, compare Walsh, Mathematische Annalen vol. 96 (1926), pp. 437–450 and Transactions of the American Mathematical Society vol. 31 (1929), pp. 477–502.
3. See Walsh, Transactions of the American Mathematical Society vol. 33 1931 The existence af a function of best approximation depends essentially on the closure of the set E.. If E contains but a tinite number of points, there are for:a given n but a tinite number of possible distributions of the orders of the poles of r n: among the points of E. For each such distribution there is loc. cit. but a single rational function of best approximation, and hence independently of this distribution there are hut a tinite number of funetions r n = of best approximation.
4. The reader may notice from the discussion which follows that in all of these cases the reasoning we give is valid or can be modified so as to be valid even if the complement of C, is finitely multiply connected. provided that C contains no isolated point.
5. Transactions of the American Mathematical Society, vol. 32 1930, pp. 794–816, and vol. 33 1931. pp. 370–388.