Proposition 23 (Fejér's theorem)

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

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

Let $n\geqslant 1$ be an integer. For every $h= (h_i)\in \mathbf{Z}^n$ , put $|h|=$ sup $_i|h_i|$ . Let $f\in \mathscr{C}(\mathbf{T}^n)$ . For every integer $N\geqslant 1$ , let us denote by $f_N$ the function on $\mathbf{T}^n$ such that

f_N(x) =\sum_{\substack{h\in\mathbf{Z}^n\\|h|\leqslant N}}\widehat{f}(h)e^{2i\pi h\cdot x}\prod_{j=1}^n\left(1-\frac{|h_j|}{N}\right)

for $x\in \mathbf{T}^n$ . Then $f_N$ converges to $f$ in $\mathscr{C}(\mathbf{T}^n)$ .

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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