^{page 123 note 1} Hille, E., Compositio Mathem., 6(1939), p. 99, p = 2.Google ScholarKober, H., Quart. J. o Math. (Oxford), 14 (1943), 49–54, referred to as K III ; p. 51.CrossRefGoogle Scholar

^{page 123 note 2} Kober, H., Bull. American, Math. Sac., 49 (1943), 437–443CrossRefGoogle Scholar, referred to as K II. Explicit formulae are given also for p = 1 and. The author's paper Bull. American Math. Soc., 48 (1942), 421–426, is referred to as K I.CrossRefGoogle Scholar

^{page 123 note 3} J. London Math. Soc., 18 (1943), 72–77. The above formulation is slightly different from the original one. A. Erdélyi gives also an explicit formula for the approximating functions, orthonormalising the sequence.Google Scholar

^{page 123 note 4} E ^p is the set of functions F(z) [z = x + iy] which, for y > 0, are regular and such that

respectively, where M = M(F,p) is independent of y ; see Hille, E. and Tamarkin, J. D., Fundamenta Mathem., 25 (1935), 329–352,CrossRefGoogle Scholar referred to as H.-T. Every F(z) of tends to a limit-function F(x) as y → 0. is the set of these limit-functions ; with “norm” | F (x)|_p [see (1.1)], and are complete normed linear spaces-for 1 ≤ p ≤ ∞, see K I, lemma 3, p. 442.

^{page 124 note 1} Riesz, F., Math. Zeitsehr, 18 (1923), 87–95.CrossRefGoogle ScholarH _p is the set of functions f(z) which, for | z | < 1, are regular and satisfy the inequality

respectively, where 0 ≦ r < 1 and M =M(f, p); is the set of the limit functions f(e ^iδ) [z → e ^iδ].

^{page 125 note 1} Math. Zeitschr, 27 (1928), 218–244, § § 13-14. There also the theory of Hilbert's operator is given, § § 17-20.CrossRefGoogle Scholar

^{page 126 note 1} K II, Lemma, p. 442 ; Lemma 3, p. 440.Google Scholar

^{page 126 note 2} H.-T., 2.1 (iii), p. 339.Google Scholar

^{page 126 note 3} This follows from Theorem 2 (a) and from Lemma 3, K II, pp. 438 and 440. Compare K III , Theorem 2, p. 54, and KI, Lemma 4, p. 423.Google Scholar

^{page 127 note 1} Interpolation and approximation in the complex domain, American Math. Soc. Coll. Publications, vol. 20, New York, 1935,Google Scholar see chapter 9, in particular 9.6. For p = ∞ the sufficiency of the condition in theorem 2 and, incidentally, (3.1), can alternatively be proved by an argument similar to that of § 6 ; but using the generalization of a theorem by Littlewood, treated in the Appendix to the present paper ; see J. London Math. Soc., 1 (1926), 229–231.Google Scholar

^{page 127 note 2} K II , Lemma 1, p. 439.Google Scholar

^{page 127 note 3} Banach, S., Théorie des opérations linéaires, Warsaw, 1932, p. 58.Google Scholar

^{page 127 note 4} Riesz, F., loc. cit., p. 89.Google Scholar

^{page 129 note 1} See footnote 2 on page 126, and KII, p. 441, footnote 15.Google Scholar

^{page 129 note 2} Compare H.-T., Theorem 2.2, p. 340.Google Scholar

^{page 130 note 1} Banach, S., loc. dt., pp. 61–65 ; is a sub-space of L ^p, (− ∞, ∞).Google Scholar

^{page 130 note 2} Compare H.-T., Theorem 2.2, p. 340.Google Scholar

^{page 130 note 3} K II, Theorem 2 (a), p. 438.Google Scholar

^{page 130 note 4} H.-T., Theorem 2.2, p. 340. There the formula is misprinted.Google Scholar

^{page 131 note 1} K I, p. 422, lines 8-9 from the bottom.Google Scholar

^{page 131 note 2} H.-T., Theorem 2.1 (ii), p. 338Google Scholar. Using the H.-T. notation, we have

As the referee has pointed out to me, the second equation of (6.7) can be deduced from the first one also this way : Replace , observing that g _ξ(z) belongs to , and take finally ξ = z.

^{page 131 note 3} Walsh, J. L., loc. cit., Chapter 2, Theorem 16.Google Scholar

^{page 132 note 1} Cf. footnote 1 on page 127.Google Scholar