“Taylor series” is to “Volterra series” as “Padé approximant” is to _________?

Question

Padé approximants are often better than Taylor series at representing a function. Given a Taylor series, one can use Wynn's epsilon algorithm to easily produce the Padé approximants to it.

Volterra series are a generalization of Taylor series that can also model "memory" phenomena. Does there exist a similar generalization of Padé approximants that can model these phenomena, or an algorithm like Wynn's to compute them from the Volterra series?

I would be at least happy to know the answer for the discrete Volterra series, which (I think) would be equivalent to something like a multivariate Padé approximant.

Originally asked at MSE, but seems too advanced for that site.

Answer 1

"I would be at least happy to know the answer for the discrete Volterra series, which (I think) would be equivalent to something like a multivariate Padé approximant."

Multi-variate versions of Padé approximants got attention in the 1970s, for example a generalization to functions of two variables was published by J.S.R. Chisolm in 1973 (see "Rational Approximates Defined from Double Power Series") and a generalization to functions of countably more variables was studied at around the same time by P.R. Graves-Morris and D.E. Roberts who were both at the University of Kent in Canterbury, leading to this generalization being called the "Canterbury approximant" (see "Calculation of Canterbury approximants").

The original papers about Canterbury approximants never seemed to get many citations, so it's easy to look through the entire list of papers that cited them to find further generalizations (for example, to functions of an uncountably infinite number of variables). Some of the most significant post-Canterbury papers on multivariate versions of Padé approximants that can be found in this list are:

• Annie Cuyt "How well can the concept of Padé approximant be generalized to the multivariate case?" (May 1999)
• Philippe Guillaume and Alain Huard. "Multivariate Padé approximation" (September 2000)

Finally, one of the original authors of the Canterbury approximant co-wrote a book with George Baker Jr. called "Padé Approximants" (first edition: 1982, second edition: 1996) which provides you with any developments which they considered significant between the aforementioned papers from the 1970s and about the time of the aforementioned papers from 1999-2000.

Finally, some steps towards a generalization to functions of complex variables can be found in my analogous question: “Taylor series” is to “Volterra series” as “Laurent series” is to _________?