Bourbaki

Remark

Tag 02JV, permanent. Cite it as [Bourbaki, Tag 02JV].

Théories spectrales, chapter II, § 1. Transformation de Fourier

In particular one recovers the determination of X(L1(Z))\mathsf{X}(L^1(\mathbf{Z})) given in Example 4 of I, p. 36.

For every integer n⩾0n\geqslant 0 and (x,y)∈Rn×Rn(x, y)\in \mathbf{R}^n\times \mathbf{R}^n, we write

x⋅y=∑j=1nxjyj x\cdot y=\sum_{j=1}^nx_jy_j

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

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Machine translation of the English, by gpt-5.4, gpt-5-6-mini, against glossary version 34. Not checked by a person. That model is a cut down version of the one the rest of this language was written by.