Geodesic
The Fourier transform of vector-valued functions
Glasgow mathematical journal, Tome 26 (1985) no. 2, pp. 181-186

Voir la notice de l'article provenant de la source Cambridge University Press

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 byform 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. 
Okada, Susumu. The Fourier transform of vector-valued functions. Glasgow mathematical journal, Tome 26 (1985) no. 2, pp. 181-186. doi: 10.1017/S0017089500005978
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