Asymptotics of minimax risk estimates of distribution density in \(L_ 2\) (Q1060788)

Let \(X_ i\), \(i\in {\mathbb{N}}\), be i.i.d. random variables with density f(X), \(f\in L_ 2[-\pi,\pi]\). The asymptotics of \[ \delta^ 2_ N({\mathcal F})=\inf_{\hat f}\sup_{f\in {\mathcal F}}E_ f\| \hat f- f\|_{L^ 2} \] are considered, when \({\mathcal F}\) is the class of densities whose Fourier coefficients belong to a parallelepiped from the space \(\ell_ 2\).