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چکیده
Exercice (6 pts) Soient ḟ et ġ deux éléments de L C (T, dθ). Vérifier que, pour tout N ∈ N, la fonction gN : θ 7−→ gN(θ) := g(Nθ) définit encore un élément de L C (T, dθ) (1 pt). Si g est un représentant de ġ (i.e une fonction mesurable 2π-périodique telle que |g|2 soit intégrable sur [0, 2π] et que g ∈ ġ), la fonction gN : θ 7−→ g(Nθ) est encore 2π-périodique car gN(θ + 2π) = g(N(θ + 2π)) = g(Nθ + 2Nπ) = g(Nθ) et de plus, on a, par la formule de changement de variable dans les intégrales et la 2π-périodicité de g : ∫
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