It seems clear to me that there should be some analogue to the notion of a region of convergence for a Volterra series.

However, it seems as though there are now many different subtle ways for things not to converge, and possibly even more so if we're talking about distributions rather than functions.

How does this work? Can Volterra series be "analytic" over some "radius" of input? Are there convergence theorems that are analogous to those for Taylor series that hold for Volterra series?


The convergence issue of a Volterra series, basically a Taylor series for functionals, is summarized as follows by Scholarpedia:

Due to its power series character, the convergence of an infinite Volterra series cannot be guaranteed for arbitrary input signals: Both the input and the output signals must be restricted to some suitable signal class. For instance, one can show that any continuous, nonlinear system can be uniformly approximated to arbitrary accuracy by a Volterra series of sufficient but finite order if the input signals are restricted to square-integrable functions on a finite interval. Although this approximation result appears to be rather general on first sight, the restriction to this type of input is quite severe. Many natural choices of input signals are precluded by this requirement such as, e.g., the standard infinite periodic forcing signals used in linear system theory.

A detailed analysis for particular classes of input signals is given in

A short answer to the question in the OP could be: the notion of a "region of convergence" does not carry over directly from Taylor series to Volterra series, because the latter is an expansion of functionals rather than functions.

  • $\begingroup$ I know it's an expansion of functionals, that's the idea behind the question - to see how more familiar concepts from real analysis generalize. But why would there be no region of convergence? Given any Volterra series, there should be a set where it converges and a set where it doesn't, right? $\endgroup$ – Mike Battaglia Sep 4 '16 at 15:48
  • $\begingroup$ the "region of convergence" would be a restriction on the set of input states, is that what you mean? Some sets of input states are considered in the linked papers. $\endgroup$ – Carlo Beenakker Sep 5 '16 at 10:59

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