Reproducing kernel resolution space and its applications

Multiple signal extraction by polynomial filtering

“…In addition to these we will also use the concepts of Hilbert resolution space which are discussed in [8], and [9], plus the concepts of Hilbert space valued random variables [7] and [10]. These latter concepts will be briefly reviewed in the following paragraphs.…”

“…While the theory associated with these concepts is again well documented in [2], [7], [10], for our purposes it will be helpful to list a few of their most important properties, namely: is Hilbert-Schmidt for all random Note that given an operator, T: H--->H, the trace of T is defined by:…”

A new performance measure for stochastic optimization in Hilbert space

“…2' 3, 7, 8 Alternatively, modified performance measures which are well defined for finitely additive error processes have been proposed for discrete time filtering problems and smoothing problems. 5 The purpose of the present paper is to formulate a new operator valued performance measure which is well defined for all finitely additive error processes and yet retains the essential characteristics of the mean squared error. Indeed, the optimization criteria derived from the proposed performance measure formally replicate the criteria derived from the mean squared error and reduce to the latter when the mean squared error is well defined.…”

“…[2][3][4][5][6][7][8] Unfortunately, this theory is characterized by a number of deficiencies which, in turn, have limited the scope of the resultant stochastic optimization theory. 2' 3 In particular, finitely additive random variable are required to model the noise phenomena encountered in "real world" stochastic optimization problems 3 but a countably additive error process is required to guarantee the existence of a well defined mean squared error.…”

Randomized system trajectories

“…Root state trajectories for the system under consideration. Thus the rather extensive theory of state in causal systems, in particular involving abstract causality in the context of resolution spaces [9] and [10] is not used. However, the system trajectories can sometimes be interpreted as state trajectories for a "dominating" system (as pointed out below), so it is entirely possible this work could be tied to the theory mentioned, and perhaps generalized in one direction with the use of resolution spaces.…”