Let {Bj,Kn,M : j = −M,...,Kn − 1} be the B-spline basis of the univariate spline space SKn,M introduced in Section 14.4. Let c > 0 be a constant specified below and set
S¯Kn,M := K n−1 j=−M ajBj,Kn,M : K n−1 j=−M |aj | ≤ c · (L + 1).
Show that if one chooses c in a suitable way then the least squares estimate
mn,(Kn,M)(·) = arg min f∈S¯Kn,M 1 n n i=1 |f(Xi) − Yi|
satisfies the bounds in Theorem 19.4 (with FK,M(L+ 1) replaced by S¯Kn,M) and in Corollary 19.1. Hint: According to de Boor (1978) there exists a constant c > 0 such that, for all aj ∈ R,
K n−1 j=−M |aj | ≤ c · sup x∈[0,1] K n−1 j=−M ajBj,Kn,M(x) .