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Sets of uniqueness for trigonometric series and integrals

Published online by Cambridge University Press:  24 October 2008

R. Henstock
R. Henstock
Affiliation:
Birkbuck CollegeLondon
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Extract

It is well known that, if a trigonometric series

converges to zero in 0 ≤ x < 2π then all the coefficients are zero. To generalize this property of the series, sets of uniqueness have been defined. A point-set E in 0 ≤ x < 2π is a set of uniqueness if every series (0·1), converging to zero in [0, 2π) − E, has zero coefficients. Otherwise E is a set of multiplicity. For example, every enumerable set is a set of uniqueness. An account of the theory may be found in Zygmund (2), chapter 11, pp. 267 et seq.

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 46 , Issue 4 , October 1950 , pp. 538 - 548
Copyright
Copyright © Cambridge Philosophical Society 1950

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References

REFERENCES

(1)Wolf, F.Contributions to a theory of summability of trigonometric integrals. (University of California Press, 1947.)Google Scholar
(2)Zygmund, A.Trigonometrical series. (Warsaw, 1935.)Google Scholar
(3)Zygmund, A.On trigonometric integrals. Ann. Math. 48 (1947), 393–440.Google Scholar
(4)Offord, A. C.Note on the uniqueness of the representation of a function by a trigonometric integral. J. London Math. Soc. 11 (1936), 171–174.CrossRefGoogle Scholar