Reproducing kernels and estimation-control duality

Consider the following two optimization problems in Hilbert spaces Hi, i=1,2: minimize ||f|| H i f ε Ui Ui = {fεHi: H i = ri (t), tψI}. We define these two problems as dual if <ψ1 (·,t), ψ1 (·,s) > H 1 = <ψ2 (·,t), ψ2 (·,s) H 2, ψt, sεI (1) r1 (t)=r2 (t), εt ε I. (2) Eq. (1) establishes an isometric isomorphism (congruence) between M1 and M2, the subspaces spanned by {ψ1 (·,t), tεI} and {ψ2 (·,t), tεI}, respectively. Together, (1) and (2) imply that the solutions f-1 and f-2 to the optimization problems are images under the congruence. Note that f-i is the projection onto Mi of any element in Ui. Thus the solution to one of the dual problems immediately gives the solution to the other, once the congruence map is established. This map can be implemented by an inner product in the reproducing kernel Hilbert space determined by R (t,s) = Hi [1]