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Lacunary entire functions

Published online by Cambridge University Press:  24 October 2008

A. C. Offord
A. C. Offord
Affiliation:
24A Norham Gardens, Oxford 0X2 6QD
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Abstract

It has been observed that lacunary functions and random functions often have many properties in common (cf. [5]). The present paper has the object of showing that lacunary entire functions behave in many respects like random entire functions. Both have the property of being large except in very small neighbourhoods of their zeros and these have a uniform angular distribution.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 114 , Issue 1 , July 1993 , pp. 67 - 83
Copyright
Copyright © Cambridge Philosophical Society 1993

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References

REFERENCES

[1]Anderson, J. M. and Clunie, J.. Entire functions of finite order and lines of Julia. Math. Zeit. 112 (1969), 59–73.CrossRefGoogle Scholar
[2]Blumethal, O.. Principe de la théorie des fonctions entière d'ordre infine (Paris, 1910).Google Scholar
[3]Cartwright, M. L.. Integral functions. Cambridge Tracts in Mathematics 44 (Cambridge, 1956).Google Scholar
[4]Davies, P. L.. Some results on the distribution of zeros of random entire functions. Proc. London Math. Soc. (3) 26 (1973), 99–141.Google Scholar
[5]Gaposhkin, W. P.. Lacanary Series and independent functions. Russian Math. Surveys 21 (1966), 1–82.CrossRefGoogle Scholar
[6]Hardy, G. H.. The zeroes of certain classes of integral functions. Part I. Proc. London Math. Soc. (2) 2 (1905), 332–9. (Collected Papers Vol. iv, 54.)CrossRefGoogle Scholar
[7]Hayman, W. K.. Angular value distribution of power series with gaps. Proc. London Math. Soc. (3) 24 (1972), 590–624.CrossRefGoogle Scholar
[8]Hayman, W. K.. The local growth of power series. Canadian Math. Bull. 17 (1974), 317–358.CrossRefGoogle Scholar
[9]Littlewood, J. E.. Collected Papers Vol. 2 (Oxford, 1982).Google Scholar
[10]Littlewood, J. E. and Offord, A. C.. On the distribution of zeros and a-values of a random integral function. II. Annals of Math. 49 (1948), 885–952CrossRefGoogle Scholar
Littlewood, J. E. and Offord, A. C.. On the distribution of zeros and a-values of a random integral function. II. Annals of Math. 50 (1948), 990–991.CrossRefGoogle Scholar
[11]Offord, A. C.. The distribution of the values of an entire function whose coefficiences are independent random variables. Proc. London Math. Soc. (3) 14A (1965), 199–238.Google Scholar
[12]Offord, A. C.. The pits property of entire functions. Journal London Math. Soc. (2) (1991), 463–475.CrossRefGoogle Scholar
[13]Polya, C.. Untersuchungen Über Lücken und Singularitaten von Potenzreihen. Math. Zeitschrift, 29 (1929), 549–640.CrossRefGoogle Scholar
[14]Valiron, O.. Lectures on the general theory of integral functions (1923; reprinted Chelsea New York, 1949).Google Scholar