6.5 Derivation of Property P6.2 Based on the Projection Theorem. Throughout this problem, we use the notation of Property P6.2 and of the Projection Theorem given in Appendix B, where H is L2. If Lk+1 =

sp{y1, . . . ,yk+1}, and Vk+1 = sp{yk+1 − ykk

+1

}, for k = 0, 1, . . . , n − 1, where ykk

+1 is the projection of yk+1 on Lk, then, Lk+1 = Lk⊕Vk+1. We assume P0 0 > 0 and R > 0.

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k+1 (a) Show the projection of xk on Ck+1, that is, z, is given by +=+Hk+1(k+1/+1), where H+1 can be determined by the orthogonality property (k+12+1) -/+1)'} = 0 Ik E{(xk - Hk+1 (k+1/+1)) (k+1 - Show Hk+1 = P'A+1 [Ak+1 P+1A+1+R] (b) Define JP'[P1], and show = k+1 I =+Je(+)-+1). k (c) Repeating the process, show +2=x+(x+-+1)+k+2(k+2+2), solving for H+2. Simplify and show +2=+Jk(+1) (d) Using induction, conclude x = +(+1+1), which yields the smoother with k=t-1.