Projection Theory Definition Given Hilbert spaces Y andZCY (Iy,)y=(y,)y,0∈z ∈Y defines the projection of y onto Z, Ily I:Y→z SMA-HPO⊙1999M Poisson in IR: Theory Practice 5
Projection Theory Property The projection Ily minimizes‖y-21|y,Vz∈Z. Why? ly-(Ily +u)y=((y-Ily)-v,(y-Ily)-vy anyz∈z Iy-IIyly-2(y-IIy, uy +la, VUEZ 0:v∈z SMA-HPO⊙1999M Poisson in IR: Theory Practice 6
Projection Theory Geometr Geometry of projection y y Orthogonality:(y-IIy, UY=0, VUEZ SMA-HPO⊙1999M Poisson in IR: Theory Practice 7
The Interpolant Theory Definition Recall Xh={∈X|vln2∈P1(Tn),WTh∈h} U∈Xh2 0 1=1 SMA-HPO⊙1999M Poisson in IR: Theory Practice 8
The Interpolant Theory Definition Given w E X, the interpolant Lh w satisfies Tn∈Xh; and th(a2)=w(c2),a=0,…,,m+1 hw(a w(i) pila m+1 SMA-HPO⊙1999M Poisson in IR: Theory Practice 9