# The Fixed-Point Theory of Strictly Causal Functions

### Eleftherios Matsikoudis and Edward A. Lee

###
EECS Department

University of California, Berkeley

Technical Report No. UCB/EECS-2013-122

June 9, 2013

### http://www.eecs.berkeley.edu/Pubs/TechRpts/2013/EECS-2013-122.pdf

We ask whether strictly causal components form well defined systems when arranged in feedback configurations. The standard interpretation for such configurations induces a fixed-point constraint on the function modelling the component involved. We define strictly causal functions formally, and show that the corresponding fixed-point problem does not always have a well defined solution. We examine the relationship between these functions and the functions that are strictly contracting with respect to a generalized distance function on signals, and argue that these strictly contracting functions are actually the functions that one ought to be interested in. We prove a constructive fixed-point theorem for these functions, introduce a corresponding induction principle, and study the related convergence process.

BibTeX citation:

@techreport{Matsikoudis:EECS-2013-122, Author = {Matsikoudis, Eleftherios and Lee, Edward A.}, Title = {The Fixed-Point Theory of Strictly Causal Functions}, Institution = {EECS Department, University of California, Berkeley}, Year = {2013}, Month = {Jun}, URL = {http://www.eecs.berkeley.edu/Pubs/TechRpts/2013/EECS-2013-122.html}, Number = {UCB/EECS-2013-122}, Abstract = {We ask whether strictly causal components form well defined systems when arranged in feedback configurations. The standard interpretation for such configurations induces a fixed-point constraint on the function modelling the component involved. We define strictly causal functions formally, and show that the corresponding fixed-point problem does not always have a well defined solution. We examine the relationship between these functions and the functions that are strictly contracting with respect to a generalized distance function on signals, and argue that these strictly contracting functions are actually the functions that one ought to be interested in. We prove a constructive fixed-point theorem for these functions, introduce a corresponding induction principle, and study the related convergence process.} }

EndNote citation:

%0 Report %A Matsikoudis, Eleftherios %A Lee, Edward A. %T The Fixed-Point Theory of Strictly Causal Functions %I EECS Department, University of California, Berkeley %D 2013 %8 June 9 %@ UCB/EECS-2013-122 %U http://www.eecs.berkeley.edu/Pubs/TechRpts/2013/EECS-2013-122.html %F Matsikoudis:EECS-2013-122