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Fix [math]\beta \ge 5/3, Q \gt 0[/math] and prove that the minimax rate of estimation over [math]\Theta(\beta, Q)[/math] with the [math]\|\cdot\|_{L_2([0,1])}[/math]-norm is given by [math]n^{-\frac{2\beta}{2\beta+1}}[/math].

Hint: Consider functions of the form

[[math]] f_j=\frac{C}{\sqrt{n}}\sum_{i=1}^N \omega_{ji} \varphi_i [[/math]]

where [math]C[/math] is a constant, [math]\omega_j \in \{0,1\}^N[/math] for some appropriately chosen [math]N[/math] and [math]\{\varphi_j\}_{j\ge 1}[/math] is the trigonometric basis.