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Non-measurable interpolation sets

II. functions regular in an angle

Published online by Cambridge University Press:  24 October 2008

M. E. Noble
M. E. Noble
Affiliation:
Queens' CollegeCambridge
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Extract

In an earlier paper the problem of the interpolation of an integral function f(z) in terms of the values taken on a sequence zn was studied by means of generalized σ-functions with zeros at zn. The purpose of the present paper is to show that the methods used in I can be modified to deal with functions of finite order and type in the angle Sα, | arg z | ≤ α. There is even a considerable saving since, for an Sα with α < π, we can escape from the awkward convergence condition Dρ (ii) of I. Most of our results, and to a great extent also their proofs, are direct generalizations of well-known results for the lattice points m + in (cf. the references at the end of I, particularly (2), (5), (6), (14), (18)).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 47 , Issue 4 , October 1951 , pp. 733 - 740
Copyright
Copyright © Cambridge Philosophical Society 1951

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References

† Noble, M. E., ‘Non-measurable interpolation sets, I, Integral functions’, Proc. Cambridge Phil. Soc. 47 (1951), 713CrossRefGoogle Scholar, referred to in future as I.

* Cartwright, M. L., Quart. J. Math. (Oxford), 7 (1936), 44–55.Google Scholar

† Ganapathy, V. Iyer, Quart. J. Math. (Oxford), 9 (1938), 206.Google Scholar

* Cf., for example, Valiron, G., Integral functions (Toulouse, 1923), p. 53.Google Scholar

† Bernstein, V., Ann. Mat. (4), 12 (1934), 173–96CrossRefGoogle Scholar, Lemma II.

* Ganapathy, V. Iyer, Quart. J. Math. (Oxford), 9 (1938), 206–15.Google Scholar

* Cf. Pflüger, A., Proc. London Math. Soc. (2), 42 (1937), 367–72.Google Scholar