Real and Complex Analysis (Rudin)

NOTES AND COMMENTS 401
orthonormal sets, M. Riesz (1926) derived this extension from a general interpolation theorem, and G.
o. Thorin (1939) discovered the complex-variable proof of M. Riesz's theorem. The proof of the text is
the Calder6n-Zygmund adaptation (1950) of Thorin's idea. Full references and discussions of other
interpolation theorems are in Chap. XII of [36].
Sec. 12.13. In slightly different form, this is in Duke Math. J., vol. 20, pp. 449-458, 1953.
Sec. 12.14. This proof is essentially that ofR. Kaufman (Math. Ann., vol. 169, p. 282,1967). E. L.
Stout (Math. Ann., vol. 177, pp. 339-340,1968) obtained a stronger result.
Chapter 13
Sec. 13.9. Runge's theorem was published in Acta Math., vol. 6, 1885. (Incidentally, this is the same
year in which the Weierstrass theorem on uniform approximation by polynomials on an interval was
published; see Mathematische Werke, vol. 3, pp. 1-37.) See [29], pp. 171-177, for a proof which is
close to the original one. The functional analysis proof of the text is known to many analysts and has
probably been independently discovered several times in recent years. It was communicated to me by
L. A. Rubel. In [13], vol. II, pp. 299-308, attention is paid to the closeness of the approximation if the
degree of the polynomial is fixed.
Exercises 5, 6. For yet another method, see D. G. Cantor, Proc. Am. Math. Soc., vol. 15, pp.
335-336,1964.
Chapter 14
General reference: [20]. Many special mapping functions are described there in great detail.
Sec. 14.3. More details on linear fractional transformations may be found in [1], pp. 22-35; in
[13], pp. 46-57; in [4]; and especially in Chap. 1 of L. R. Ford's book "Automorphic Functions,"
McGraw-Hili Book Company, New York, 1929.
Sec. 14.5. Normal families were introduced by Montel. See Chap. 15 of [13].
Sec. 14.8. The history of Riemann's theorem is discussed in [13], pp. 32{}-321, and in [29],
p. 230. Koebe's proof (Exercise 26) is in J. fUr reine und angew. Math., vol. 145, pp. 177-223, 1915;
doubly connected regions are also considered there.
Sec. 14.14. Much more is true than just I a 2 1 S; 2: In fact, I an I s; n for all n ~ 2. This was conjectured
by Bieberbach in 1916, and proved by L. de Branges in 1984 [Acta Math., vol. 154, pp. 137-152,
(1985)]. Moreover, if I an I = n for just one n ~ 2, then f is one of the Koebe functions of Example
14.11.
Sec. 14.19. The boundary behavior of conformal mappings was investigated by CarathCodory in
Math. Ann., vol. 73, pp. 323-370, 1913. Theorem 14.19 was proved there for regions bounded by
Jordan curves, and the notion of prime ends was introduced. See also [4], vol. II, pp. 88-107.
Exercise 24. This proof is due to Y. N. Moschovakis.
Chapter 15
Sec. 15.9. The relation between canonical products and entire functions of finite order is discussed in
Chap. 2 of [3], Chap. VII of [29], and Chap. VIII of [31].
Sec. 15.25. See Szasz, Math. Ann., vol. 77, pp. 482-496, 1916, for further results in this direction.
Also Chap. II of [21].
Chapter 16
The classical work on Riemann surfaces is [32]. (The first edition appeared in 1913.) Other references:
Chap. VI of [1], Chap. 10 of [13], Chap. VI of [29], and [30].
Sec. 16.5. Ostrowski's theorem is in J. London Math. Soc., vol. 1, pp. 251-263, 1926. See J.-P.
Kahane, Lacunary Taylor and Fourier Series, Bull. Am. Math. Soc., vol. 70, pp. 199-213, 1964, for a
more recent account of gap series.
Sec. 16.15. The approach to the monodromy theorem was a little simpler in the first edition of
this book. It used the fact that every simply connected plane region is homeomorphic to a convex