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?



Do Volterra series have a region of convergence, as do Taylor series?
The convergence issue of a Volterra series, basically a Taylor series for *functionals*, is summarized as follows by [Scholarpedia:](http://www.scholarpedia.org/article/Volterra_and_Wiener_series)



> 
> 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 


* [On
the convergence of Volterra series of finite dimensional quadratic
MIMO (multiple-input-multiple-output) systems.](http://recherche.ircam.fr/anasyn/helie/PDF/IJC_Helie_Laroche_perso.pdf)
* [Analysis of the Volterra Series Convergence](http://link.springer.com/chapter/10.1007%2F978-94-011-4229-8_26)
* [A New Convergence Criteria of Volterra Series for Harmonic Inputs](https://sem.org/wp-content/uploads/2016/01/sem.org-IMAC-XVI-16th-Int-162203-A-New-Convergence-Criteria-Volterra-Series-Harmonic-Inputs.pdf)


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.


