On a class of ill-posed estimation problems and a related gradient iteration

This paper deals with the application of the gradient iteration to a class of ill-posed estimation problems arising in many different contexts, such as system and channel identification, radar mapping and resolution, enhancement or restoration of optical images, and so on. The basic problem is one of infinite-dimensional linear regression type where the unknown
\chi
is a function constrained in an arbitrary functional Hilbert space
b
, and the observation noise is a second-order stochastic process. It is shown that a necessary and sufficient condition for the gradient iteration to define a sequence of estimates
\{\hat{x}_{p}\}
in the constraint space
b
is one of strong stochastic nonsingularity for a hypothetical detection problem. Conditions that guarantee the convergence of the gradient iterates
\{\hat{x}_{p}\}
in a suitable sense are also given.