# Volterra series

integro-power series

A series containing the powers of the unknown function under the integral sign. Let $K ( s , t _ {1} \dots t _ {k} )$ be a continuous function in all variables in a cube $[ a , b ] ^ {k+} 1$ and let $U ( s)$ be an arbitrary continuous function on $[ a , b ]$. The expression

$$U ^ {\alpha _ {0} } ( s) \int\limits _ { a } ^ { b } \dots \int\limits _ { a } ^ { b } K ( s , t _ {1} \dots t _ {k} ) U ^ {\alpha _ {1} } ( t _ {1} ) \dots U ^ {\alpha _ {k} } ( t _ {k} ) d t _ {1} \dots d t _ {k} ,$$

where $\alpha _ {0} \dots \alpha _ {k}$ are non-negative integers and $\alpha _ {0} + \dots + \alpha _ {k} = m$, is called a Volterra term of degree $m$ in $U$. Two Volterra terms of degree $m$ belong to the same type if they differ only in their kernels $K$. The finite sum of Volterra terms (of all types) of degree $m$ is called a Volterra form of degree $m$ in the function $U$. It is denoted by

$$W _ {m} \left ( \begin{array}{c} s \\ U \end{array} \right ) .$$

Let

$$| W | _ {m} \left ( \begin{array}{c} s \\ U \end{array} \right )$$

denote the Volterra form in which the kernel $K$ is replaced by $| K |$, and let

$$\widetilde{U} = \ \max _ {[ a , b ] } | U ( s) | ,\ \ \widetilde{W} _ {m} = \ \max _ {[ a , b ] } | W | _ {m} \left ( \begin{array}{c} s \\ U \end{array} \right ) ;$$

then

$$\left | W _ {m} \left ( \begin{array}{c} s \\ U \end{array} \right ) \right | \leq \ \widetilde{W} _ {m} \widetilde{U} {} ^ {m} .$$

The expression

$$W _ {0} \left ( \begin{array}{c} s \\ U \end{array} \right ) + W _ {1} \left ( \begin{array}{c} s \\ U \end{array} \right ) + W _ {2} \left ( \begin{array}{c} s \\ U \end{array} \right ) + \dots$$

is called a Volterra series. If the series of numbers $\widetilde{W} _ {0} + \widetilde{W} _ {1} \widetilde{U} + \widetilde{W} _ {2} \widetilde{U} {} ^ {2} + \dots$ 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 $[ a , b ]$.

Analogously one introduces Volterra series in several functional arguments, and Volterra series in which $[ a , b ]$ 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.

#### References

 [1] A.M. Lyapunov, "On equilibrium figures deviating slightly from ellipsoids of rotation of homogeneous fluid masses" , Collected Works , 4 , Moscow (1959) (In Russian) [2] E. Schmidt, "Zur Theorie der linearen und nichtlinearen Integralgleichungen III" Math. Ann. , 65 (1908) pp. 370–399 [3] M.M. Vainberg, V.A. Trenogin, "Theory of branching of solutions of non-linear equations" , Noordhoff (1974) (Translated from Russian)

A non-linear input-output dynamical system with input $u$ and output $y$ gives rise to a Volterra series of the form

$$y( t) = \int\limits _ {- \infty } ^ { {+ } \infty } h _ {1} ( \tau _ {1} ) u ( t - \tau _ {1} ) d \tau _ {1} +$$

$$+ \int\limits _ {- \infty } ^ { {+ } \infty } \int\limits _ {- \infty } ^ { {+ } \infty } h _ {2} ( \tau _ {1} , \tau _ {2} ) u ( t - \tau _ {1} ) u( t - \tau _ {2} ) d \tau _ {1} d \tau _ {2} + \dots +$$

$$+ \int\limits _ {- \infty } ^ { {+ } \infty } \dots \int\limits _ {- \infty } ^ { {+ } \infty } h _ {n} ( \tau _ {1} \dots \tau _ {n} ) u( t- \tau _ {1} ) \dots u ( t - \tau _ {n} )$$

$$d \tau _ {1} \dots d \tau _ {n} + \dots ,$$

in which $h _ {n} ( \tau _ {1} \dots \tau _ {n} ) = 0$ if $\tau _ {j} < 0$ for some $j$. Such series were first introduced by V. Volterra, [a1], and first applied to questions of system theory by N. Wiener, leading to Wiener integrals, [a2]. Cf. [a3] for an extensive discussion of Volterra series in system theory.

#### References

 [a1] V. Volterra, "Theory of functionals and of integral and integro-differential equations" , Dover, reprint (1959) (Translated from French) [a2] N. Wiener, "Nonlinear problems in random theory" , M.I.T. (1958) [a3] M. Schetzen, "The Volterra and Wiener theories of nonlinear systems" , Wiley (1980)
How to Cite This Entry:
Volterra series. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Volterra_series&oldid=49159
This article was adapted from an original article by V.A. Trenogin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article