Derive the constraints on the j in the thin-plate spline expansion

(5.39) to guarantee that the penalty J(f) is finite. How else could one

ensure that the penalty was finite?

The solution has the form N f{I) = :81] -I- TI + Z jhj{1)1 (5.39) j=1 where fig-(:5) 2 H3: :cj||21og|]:r :ch. These hj are examples of radial basis functions, which are discussed in more detail in the next section. The coeicients are found by plugging (5.39) into [5.37), which reduces to a nitadimensional penalised least squares problem. For the penalty,r to be nite} the coefcients er- have to satisfy a set of linear constraints; see Exercise 5.14