> 0}

A regularized inverse of R is in fact a family of operators { R;;- 1, a > 0} such that each R;; 1 : L2(/Lf[}) ----+ L 2(JLII}) is bounded and such that for each f E L2(/Lif}):

I 'PI.n(t):=et+t' t>O, n>O, (2.3)

1 'P2o(t) := l lin x)(t), f > 0, Ct > 0. (2.4)

'f!l.o(R) = (o1 + R)- 1, 0 > 0. (2.5) If R is obtained from preconditioning with f = K* the Moore-Penrose inverse yields an interesting interpretation as a penalized least-squares solution, since one can show that

where ?JK is the range of K. The second family in Eq. (2.4) yields the socalled spectral cut-off type inverse of R. In this chapter we will be exclusively dealing with the second family and throughout employ the inverse

Q > 0, (2.7) without further reference.