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''integro-power series''
 
''integro-power series''
  
A series containing the powers of the unknown function under the integral sign. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096870/v0968701.png" /> be a continuous function in all variables in a cube <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096870/v0968702.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096870/v0968703.png" /> be an arbitrary continuous function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096870/v0968704.png" />. The expression
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096870/v0968705.png" /></td> </tr></table>
+
$$
 +
U ^ {\alpha _ {0} } ( s)
 +
\int\limits _ { a } ^ { b }  \dots \int\limits _ { a } ^ { b }
 +
K ( s , t _ {1} \dots t _ {k} )
 +
U ^ {\alpha _ {1} } ( t _ {1} ) \dots
 +
U ^ {\alpha _ {k} } ( t _ {k} )
 +
d t _ {1} \dots d t _ {k} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096870/v0968706.png" /> are non-negative integers and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096870/v0968707.png" />, is called a Volterra term of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096870/v0968708.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096870/v0968709.png" />. Two Volterra terms of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096870/v09687010.png" /> belong to the same type if they differ only in their kernels <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096870/v09687011.png" />. The finite sum of Volterra terms (of all types) of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096870/v09687012.png" /> is called a Volterra form of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096870/v09687013.png" /> in the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096870/v09687014.png" />. It is denoted by
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096870/v09687015.png" /></td> </tr></table>
+
$$
 +
W _ {m} \left ( \begin{array}{c}
 +
s \\
 +
U
 +
\end{array}
 +
\right ) .
 +
$$
  
 
Let
 
Let
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096870/v09687016.png" /></td> </tr></table>
+
$$
 +
| W | _ {m} \left ( \begin{array}{c}
 +
s \\
 +
U
 +
\end{array}
 +
\right )
 +
$$
 +
 
 +
denote the Volterra form in which the kernel  $  K $
 +
is replaced by  $  | K | $,
 +
and let
  
denote the Volterra form in which the kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096870/v09687017.png" /> is replaced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096870/v09687018.png" />, and let
+
$$
 +
\widetilde{U}  = \
 +
\max _ {[ a , b ] }  | U ( s) | ,\ \
 +
\widetilde{W}  _ {m}  = \
 +
\max _ {[ a , b ] }  | W | _ {m} \left ( \begin{array}{c}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096870/v09687019.png" /></td> </tr></table>
+
s \\
 +
U
 +
\end{array}
 +
\right ) ;
 +
$$
  
 
then
 
then
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096870/v09687020.png" /></td> </tr></table>
+
$$
 +
\left | W _ {m} \left ( \begin{array}{c}
 +
s \\
 +
U
 +
\end{array}
 +
\right ) \right |  \leq  \
 +
\widetilde{W}  _ {m} \widetilde{U}  {}  ^ {m} .
 +
$$
  
 
The expression
 
The expression
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096870/v09687021.png" /></td> </tr></table>
+
$$
 +
W _ {0} \left ( \begin{array}{c}
 +
s \\
 +
U
 +
\end{array}
 +
\right ) +
 +
W _ {1} \left ( \begin{array}{c}
 +
s \\
 +
U
 +
\end{array}
 +
\right ) +
 +
W _ {2} \left ( \begin{array}{c}
 +
s \\
 +
U
 +
\end{array}
 +
\right ) + \dots
 +
$$
  
is called a Volterra series. If the series of numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096870/v09687022.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096870/v09687023.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096870/v09687024.png" /> 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|power series]].
+
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|power series]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.M. Lyapunov,  "On equilibrium figures deviating slightly from ellipsoids of rotation of homogeneous fluid masses" , ''Collected Works'' , '''4''' , Moscow  (1959)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E. Schmidt,  "Zur Theorie der linearen und nichtlinearen Integralgleichungen III"  ''Math. Ann.'' , '''65'''  (1908)  pp. 370–399</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M.M. Vainberg,  V.A. Trenogin,  "Theory of branching of solutions of non-linear equations" , Noordhoff  (1974)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.M. Lyapunov,  "On equilibrium figures deviating slightly from ellipsoids of rotation of homogeneous fluid masses" , ''Collected Works'' , '''4''' , Moscow  (1959)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E. Schmidt,  "Zur Theorie der linearen und nichtlinearen Integralgleichungen III"  ''Math. Ann.'' , '''65'''  (1908)  pp. 370–399</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M.M. Vainberg,  V.A. Trenogin,  "Theory of branching of solutions of non-linear equations" , Noordhoff  (1974)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
A non-linear input-output dynamical system with input <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096870/v09687025.png" /> and output <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096870/v09687026.png" /> gives rise to a Volterra series of the form
+
A non-linear input-output dynamical system with input $  u $
 +
and output $  y $
 +
gives rise to a Volterra series of the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096870/v09687027.png" /></td> </tr></table>
+
$$
 +
y( t)  = \int\limits _ {- \infty } ^ { {+ }  \infty }
 +
h _ {1} ( \tau _ {1} ) u ( t - \tau _ {1} )  d \tau _ {1} +
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096870/v09687028.png" /></td> </tr></table>
+
$$
 +
+
 +
\int\limits _ {- \infty } ^ { {+ }  \infty } \int\limits _ {- \infty } ^ { {+ }  \infty } h _ {2} ( \tau _ {1} , \tau _ {2} ) u ( t - \tau _ {1} ) u(
 +
t - \tau _ {2} )  d \tau _ {1}  d \tau _ {2} + \dots +
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096870/v09687029.png" /></td> </tr></table>
+
$$
 +
+
 +
\int\limits _ {- \infty } ^ { {+ }  \infty } \dots \int\limits _ {- \infty } ^ { {+ }  \infty } h _ {n} ( \tau _ {1} \dots \tau _ {n} ) u( t- \tau _ {1} ) \dots u ( t - \tau _ {n} )
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096870/v09687030.png" /></td> </tr></table>
+
$$
 +
d \tau _ {1} \dots d \tau _ {n} + \dots ,
 +
$$
  
in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096870/v09687031.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096870/v09687032.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096870/v09687033.png" />. Such series were first introduced by V. Volterra, [[#References|[a1]]], and first applied to questions of system theory by N. Wiener, leading to Wiener integrals, [[#References|[a2]]]. Cf. [[#References|[a3]]] for an extensive discussion of Volterra series in system theory.
+
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, [[#References|[a1]]], and first applied to questions of system theory by N. Wiener, leading to Wiener integrals, [[#References|[a2]]]. Cf. [[#References|[a3]]] for an extensive discussion of Volterra series in system theory.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  V. Volterra,  "Theory of functionals and of integral and integro-differential equations" , Dover, reprint  (1959)  (Translated from French)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  N. Wiener,  "Nonlinear problems in random theory" , M.I.T.  (1958)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M. Schetzen,  "The Volterra and Wiener theories of nonlinear systems" , Wiley  (1980)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  V. Volterra,  "Theory of functionals and of integral and integro-differential equations" , Dover, reprint  (1959)  (Translated from French)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  N. Wiener,  "Nonlinear problems in random theory" , M.I.T.  (1958)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M. Schetzen,  "The Volterra and Wiener theories of nonlinear systems" , Wiley  (1980)</TD></TR></table>

Latest revision as of 08:28, 6 June 2020


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)

Comments

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=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