## Volterra series

### Richard Garner – 26 July 2023

A function R --> R may encode a system whose output at a given time depends only on its input at that time. A natural generalisation involves shift-invariant operators l^2(R) --> l^2(R), which may encode systems whose output at a given time depends on its input at all preceding times. One can reasonably ask what happens to the differential calculus in this situation. Well: ordinary derivatives are replaced by Frechet derivatives; and Taylor series are replaced by so-called Volterra series.

This talk will sketch the general theory with a vagueness that would
make a functional analyst wince; the point really being to provide grist
for a category theorist who fancied turning the whole business into
combinatorics or profunctor calculus.

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