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A mini-gap theorem for Fourier series

Published online by Cambridge University Press:  24 October 2008

W. K. Hayman
W. K. Hayman
Affiliation:
Imperial College, London
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Extract

Suppose that

belongs to L2( − π, π). If most of the coefficients vanish then f (x) cannot be too small in a certain interval without being small generally. More precisely Ingham ((2), Theorem 1) has proved the following

THEOREM A. Suppose that f (x) is given by (1·1) and that an = 0, except for a sequence n = nν, where nν+1 − nν ≥ C. Then given ∈ > 0 there exists a constant A (∈), such that we have for any real x1

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 64 , Issue 1 , January 1968 , pp. 61 - 66
Copyright
Copyright © Cambridge Philosophical Society 1968

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References

REFERENCES

(1)Hayman, W. K.Mean p -valent functions with mini-gaps. Math. Nachr. (to be published).Google Scholar
(2)Ingham, A. E.Some trigonometrical inequalities with applications to the theory of series. Math. Z. 41 (1936), 367–379.CrossRefGoogle Scholar
(3)Kadec, M. I.The exact value of the Paley–Wiener constant (Russian). Dokl. Akad. Nauk SSSR 155 (1964), 1253–1254.Google Scholar
(4)Levinson, N.On non-harmonic Fourier series. Ann. of Math. 37 (1936), 919–936.CrossRefGoogle Scholar
(5)Paley, R. E. A. C. and Wiener, N.Fourier Transforms in the Complex domain, volume xix (Amer. Math. Soc. Colloquium publications, 1934).Google Scholar