Functional Equations in the Theory of Dynamic Programming. v. Positivity and Quasi-linearity.

10 This theorem generalizes a result proved by 0. Lokki, "Uber das Randwertproblem der analytischen Funktionen," Ann. Acad. Sci. Fennicae, Ser. A-I, 144, 1-8, 1952. 11 Under the present assumptions, the result for harmonic functions corresponding to Theorem 3 was conjectured by W. Kaplan in a conversation with one of the present authors; a proof of it was obtained jointly, and the result appears in W. Kaplan, "Approximation by Entire Functions," Mich. Math. J., 3, 1955 (in press). 12 This theorem extends a result due to Lusin, who showed that if X(e) is finite almost everywhere, and measurable, then there exists a trigonometric series in 0 that is both Abel and Riemann summable to X(0) almost everywhere. It is not possible to replace "Abel" by "Riemann" in our theorem, as follows from a result proved by Germeier. Cf. N. N. Lusin, Integral and Trigonometric Series (in Russiana), (Moscow and Leningrad, 1951), pp. 236, 488 ff. I3 If, in particular, B, is assumed to have continuous curvature, then J, may be taken along the inner normal to B, at co.