Exercise 16, § 6 Integration, IV, p. 139 Let μ\muμ be Lebesgue measure on X=[0,2π]X = [0, 2\pi]X=[0,2π]. The sequence of functions fnf_nfn such that f0(x)=12π,f2n−1(x)=1πcosnx,f2n(x)=1πsinnx(n≥1) f_0(x) = \frac{1}{\sqrt{2\pi}}, \quad f_{2n-1}(x) = \frac{1}{\sqrt{\pi}} \cos nx, \quad f_{2n}(x) = \frac{1}{\sqrt{\pi}} \sin nx \quad (n \geq 1) f0(x)=2π1,f2n−1(x)=π1cosnx,f2n(x)=π1sinnx(n≥1) is an orthonormal and total sequence in the space LC2\mathscr{L}_C^2LC2 (cf. GT, X, §4, Prop. 8). Show that the corresponding Lebesgue function Hn(x)H_n(x)Hn(x) is independent of xxx and that Hn∼4/πlognH_n \sim 4/\pi \log nHn∼4/πlogn. Solution No solution has been written yet.