# Volterra series

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integro-power series

A series containing the powers of the unknown function under the integral sign. Let be a continuous function in all variables in a cube and let be an arbitrary continuous function on . The expression where are non-negative integers and , is called a Volterra term of degree in . Two Volterra terms of degree belong to the same type if they differ only in their kernels . The finite sum of Volterra terms (of all types) of degree is called a Volterra form of degree in the function . It is denoted by Let denote the Volterra form in which the kernel is replaced by , and let then The expression is called a Volterra series. If the series of numbers converges, then the Volterra series is called regularly convergent. In this case the Volterra series converges absolutely and uniformly, and its sum is continuous on .

Analogously one introduces Volterra series in several functional arguments, and Volterra series in which is replaced by some closed bounded set in a finite-dimensional Euclidean space. Volterra series are a particular case of the more general concept of an abstract power series.

How to Cite This Entry:
Volterra series. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Volterra_series&oldid=11667
This article was adapted from an original article by V.A. Trenogin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article