Let L > 0 be arbitrary. Let F(d) K,M be the set of all multivariate piecewise polynomials of degree M (or less, in each coordinate) with respect to an equidistant partition of [0, 1]d into Kd cubes,...

Let L > 0 be arbitrary. Let F(d) K,M be the set of all multivariate piecewise polynomials of degree M (or less, in each coordinate) with respect to an equidistant partition of [0, 1]d into Kd cubes, set

                              F(d) K,M(L + 1) =  f ∈ F(d) K,M : sup x∈[0,1]d |f(x)| ≤ L + 1

and define the estimate mn,(K,M) by

                              mn,(K,M) = arg min f∈F(d) K,M(L+1) 1 n n i=1 |f(Xi) − Yi| 2 .

Show that there exists a constant c depending only on L and d such that

                          E  |mn,(K,M)(x) − m(x)| 2 µ(dx)

                        ≤ c · (M + 1)dKd n + 2 inf f∈F(d) K,M(L+1)

for all distributions of (X, Y ) with X ∈ [0, 1]d a.s. and |Y | ≤ L a.s. Hint: Proceed as in the proof of Theorem 19.4.