Let (u(x)=mathbb{1}_{[0,1 / 3)}(x)). Prove that the Haar-Fourier series diverges at (x=frac{1}{3}). [ verify that (lim inf

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Let \(u(x)=\mathbb{1}_{[0,1 / 3)}(x)\). Prove that the Haar-Fourier series diverges at \(x=\frac{1}{3}\).

[ verify that \(\lim \inf _{N ightarrow \infty} s_{N}\left(u, \frac{1}{3}ight)<\lim \sup _{N ightarrow \infty} s_{N}\left(u, \frac{1}{3}ight)\).]

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