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The Fourier transform of vector-valued functions

Published online by Cambridge University Press:  18 May 2009

Susumu Okada
Susumu Okada
Affiliation:
Department of Mathematical Sciences, San Diego State University, San Diego, CA 92182-0314, U.S.A.
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For each natural number n, let un(x)=(1—cos nx)/πnx2(xɛℝ). It is well–known that a bounded continuous function f on the real line ℝ is the Fourier transform of an integrable function on ℝ if and only if the functions Φn(f) (n= 1, 2,…), defined by

form a Cauchy sequence in the space L1(ℝ) (cf. [2]). Such a characterization, which can be extended to functions defined on a locally compact Abelian group more general than ℝ, is based on the fact that the space L1(ℝ) is complete with respect to convergence in mean.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 26 , Issue 2 , July 1985 , pp. 181 - 186
Copyright
Copyright © Glasgow Mathematical Journal Trust 1985

References

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