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On uniform interpolation sets

Published online by Cambridge University Press:  24 October 2008

J. P. Earl
J. P. Earl
Affiliation:
University of Kent
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Extract

A well-known result in the interpolation theory of integral functions (see Whittaker (16, 17), Pólya (14), Iyer (2), Pfluger (10)) states that an integral function of at most type K < ½π of order 2 bounded at the lattice points m + in (m, n = 0, ± 1, ± 2, … ) is necessarily constant. That the value ½π cannot be increased is shown by the Weierstrass σ-function. The result has, however, been generalized in several ways; the lattice points being replaced by more general sets and the bounded-ness condition by one of restricted rate of growth.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 62 , Issue 4 , October 1966 , pp. 721 - 742
Copyright
Copyright © Cambridge Philosophical Society 1966

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References

REFERENCES

(1)Cartwright, M. L.On functions bounded at the lattice points in an angle. Proc. London Math. Soc. (2), 43 (1937), 26–32.Google Scholar
(2)Iyer, V. G.A note on integral functions of order 2 bounded at the lattice points. J. London Math. Soc. 11 (1936), 247–9.CrossRefGoogle Scholar
(3)Iyer, V. G.On effective sets of points in relation to integral functions. Trans. American Math. Soc. 42 (1937), 358–65.CrossRefGoogle Scholar
(4)Iyer, V. G. (Correction to (3).) Trans. American Math. Soc. 43 (1938), 494.Google Scholar
(5)Iyer, V. G.Some theorems on functions regular in an angle. Quart. J. Math. (Oxford) 9 (1938), 206–15.CrossRefGoogle Scholar
(6)Iyer, V. G.Determinative sets of points for classes of integral functions. Math. Z. 44 (1939), 195–200.CrossRefGoogle Scholar
(7)Kershner, R.The number of circles covering a set. American J. Math. 61 (1939), 665–71.CrossRefGoogle Scholar
(8)Noble, M. E.Non-measurable interpolation sets. I. Proc. Cambridge Philos. Soc. 47 (1951), 713–32.CrossRefGoogle Scholar
(9)Noble, M. E.Non-measurable interpolation sets. II. Proc. Cambridge Philos. Soc. 47 (1951), 733–40.CrossRefGoogle Scholar
(10)Pfluger, A.On analytic functions bounded at the lattice points. Proc. London Mach. Soc. (2), 42 (1937), 305–15.CrossRefGoogle Scholar
(11)Pfluger, A.Über Interpolation ganzer Funktionen. Comment. Math. Hel. 14 (1941–1942), 314–49.CrossRefGoogle Scholar
(12)Pfluger, A.Über ganze Funktionen ganzer Ordnung. Comment. Math. Hel. 18 (1945–1946), 177–203.CrossRefGoogle Scholar
(13)Pólya, G.Untersuchungen über Lücken und Singularitäten von Potenzreihen. Math. Z. 29 (1929), 549–640.CrossRefGoogle Scholar
(14)Pólya, G.Bemerkung zu der Lösung der Aufgabe 105. Jber. Deutsch. Math. Verein 43 (1933), 67–69.Google Scholar
(15)Turan, P.On a property of lacunary power series. Acta Sci. Math. Szeged 14 (1952), 209–18.Google Scholar
(16)Whittaker, J. M.On the flat regions of integral functions of finite order. Proc. Edinburgh Math. Soc. 2 (1930), 111–28.CrossRefGoogle Scholar
(17)Whittaker, J. M.On the fluctuation of integral and meromorphic functions. Proc. London Math. Soc. (2), 37 (1934), 383–401.CrossRefGoogle Scholar