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On a theorem of Zygmund

Published online by Cambridge University Press:  24 October 2008

M. Kac
M. Kac
Affiliation:
Cornell University, Ithaca, New York
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Extract

Zygmund (1) proved in 1936 that if f(x) ∈Lp(−∞, ∞) (1 ≤ p ≤ 2), the Fourier transform

exists for almost every λ, in the sense that

exists for almost every λ. The purpose of this note is to provide a short proof of this theorem.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 47 , Issue 3 , July 1951 , pp. 475 - 476
Copyright
Copyright © Cambridge Philosophical Society 1951

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References

REFERENCES

(1)Zygmund, A.A remark on Fourier transforms. Proc. Cambridge Phil. Soc. 32 (1936), 321–7.CrossRefGoogle Scholar
(2)Kac, M.Convergence and divergence of non-harmonic gap series. Duke Math. J. 8 (1941), 541–5.CrossRefGoogle Scholar
(3)Hartman, P.Divergence of non-harmonic gap series. Duke Math. J. 9 (1942), 404–5.CrossRefGoogle Scholar
(4)Menchoff, D.Sur les séries de fonctions orthogonales. Fund. Math. 11 (1928), 375–420, Theorem 12.Google Scholar