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A series of the form
 
A 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/n/n066/n066440/n0664401.png" /></td> </tr></table>
+
$$
 +
\sum _ { n= } 0 ^  \infty 
 +
a _ {n} J _ {\nu + n }  ( z) ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066440/n0664402.png" /> is the Bessel function (cylinder function of the first kind, cf. [[Bessel functions|Bessel functions]]) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066440/n0664403.png" /> is a (real or complex) number. C.G. Neumann
+
where $  J _ {\nu + n }  $
 +
is the Bessel function (cylinder function of the first kind, cf. [[Bessel functions|Bessel functions]]) and $  \nu $
 +
is a (real or complex) number. C.G. Neumann
  
considered the special case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066440/n0664404.png" /> is an integer. He showed that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066440/n0664405.png" /> is an analytic function in a closed disc with centre at the coordinate origin, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066440/n0664406.png" /> is an interior point and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066440/n0664407.png" /> denotes the boundary of the disc, then
+
considered the special case when $  \nu $
 +
is an integer. He showed that if $  f ( z) $
 +
is an analytic function in a closed disc with centre at the coordinate origin, $  z $
 +
is an interior point and $  C $
 +
denotes the boundary of the disc, 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/n/n066/n066440/n0664408.png" /></td> </tr></table>
+
$$
 +
f ( z)  = \
 +
\sum _ { n= } 0 ^  \infty 
 +
a _ {n} J _ {n} ( z) ,
 +
$$
  
 
where
 
where
  
<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/n/n066/n066440/n0664409.png" /></td> </tr></table>
+
$$
 +
a _ {0}  = f ( 0) ,\ \
 +
a _ {n}  =
 +
\frac{1}{\pi i }
  
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066440/n06644010.png" /> is a polynomial of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066440/n06644011.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066440/n06644012.png" />:
+
\int\limits _ { C }
 +
O _ {n} ( t) f ( t)  dt
 +
$$
  
<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/n/n066/n066440/n06644013.png" /></td> </tr></table>
+
and  $  O _ {n} $
 +
is a polynomial of degree  $  n+ 1 $
 +
in  $  1 / t $:
  
<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/n/n066/n066440/n06644014.png" /></td> </tr></table>
+
$$
 +
O _ {0} ( t)  =
 +
\frac{1}{t}
 +
,
 +
$$
  
<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/n/n066/n066440/n06644015.png" /></td> </tr></table>
+
$$
 +
O _ {n} ( t)  =
 +
\frac{1}{2 t  ^ {n+} 1 }
 +
\times
 +
$$
  
it is usually called the Neumann polynomial of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066440/n06644016.png" />. (Neumann himself called it a Bessel function of second order. Nowadays this term is used to denote one of the solutions of the Bessel equation.) Examples of the representation of functions by means of a Neumann series:
+
$$
 +
\times
 +
\int\limits _ { 0 } ^  \infty  e  ^ {-} x [ ( x + \sqrt {x  ^ {2} + t  ^ {2} } )  ^ {n}
 +
+ ( x - \sqrt {x  ^ {2} + t  ^ {2} } ) ^ {n} ]  dx ,\  n \geq  1 ;
 +
$$
  
<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/n/n066/n066440/n06644017.png" /></td> </tr></table>
+
it is usually called the Neumann polynomial of order  $  n $.  
 +
(Neumann himself called it a Bessel function of second order. Nowadays this term is used to denote one of the solutions of the Bessel equation.) Examples of the representation of functions by means of a Neumann series:
  
<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/n/n066/n066440/n06644018.png" /></td> </tr></table>
+
$$
 +
\cos  ( z  \sin  \phi )  = \
 +
J _ {0} ( z) +
 +
2 \sum _ { n= } 1 ^  \infty 
 +
J _ {2n} ( z)  \cos  2 n \phi ,
 +
$$
  
<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/n/n066/n066440/n06644019.png" /></td> </tr></table>
+
$$
 +
\sin  ( z  \sin  \phi )  = 2 \sum _ { n= } 1 ^  \infty  J _ {2n-} 1 ( z)  \sin  ( 2n - 1 ) \phi ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066440/n06644020.png" /> is an arbitrary number not equal to a non-negative integer and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066440/n06644021.png" /> is the [[Gamma-function|gamma-function]].
+
$$
 +
\left ( {
 +
\frac{z}{2}
 +
} \right )  ^  \mu  = \sum _ { n= } 0 ^  \infty 
 +
\frac{( \mu + 2n ) \Gamma ( \mu + n ) }{n ! }
 +
J _ {\mu + 2n }  ( z) ,
 +
$$
 +
 
 +
where  $  \mu $
 +
is an arbitrary number not equal to a non-negative integer and $  \Gamma $
 +
is the [[Gamma-function|gamma-function]].
  
 
In the theory of Fredholm integral equations (cf. [[Fredholm equation|Fredholm equation]])
 
In the theory of Fredholm integral equations (cf. [[Fredholm equation|Fredholm equation]])
  
<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/n/n066/n066440/n06644022.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
\phi ( x) - \lambda \int\limits _ { a } ^ { b }
 +
K ( x , s ) \phi ( s)  ds  = \
 +
f ( x) ,\ \
 +
x \in [ a , b ] ,
 +
$$
  
a Neumann series is defined as the expansion of the resolvent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066440/n06644023.png" /> of the kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066440/n06644024.png" />:
+
a Neumann series is defined as the expansion of the resolvent $  R ( x , s ;  \lambda ) $
 +
of the kernel $  K $:
  
<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/n/n066/n066440/n06644025.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
R ( x , s ; \lambda )  = \
 +
\sum _ { n= } 1 ^  \infty 
 +
\lambda  ^ {n} K _ {n} ( x , s ) ,
 +
$$
  
where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066440/n06644026.png" /> are the iterated kernels (of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066440/n06644027.png" />), which are defined by the recurrence formulas
+
where the $  K _ {n} $
 +
are the iterated kernels (of $  K $),  
 +
which are defined by the recurrence formulas
  
<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/n/n066/n066440/n06644028.png" /></td> </tr></table>
+
$$
 +
K _ {1} ( x , s )  = K ( x , s ) ,
 +
$$
  
<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/n/n066/n066440/n06644029.png" /></td> </tr></table>
+
$$
 +
K _ {n} ( x , s )  = \int\limits _ { a } ^ { b }  K _ {n-} 1 ( x , t ) K ( t , s )  dt ,\  n \geq  2 .
 +
$$
  
By means of (2) the solution of (1) for small <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066440/n06644030.png" /> can be represented by
+
By means of (2) the solution of (1) for small $  \lambda $
 +
can be represented 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/n/n066/n066440/n06644031.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{3 }
 +
\phi ( x)  = f ( x) + \sum _ { k= } 1 ^  \infty  \lambda  ^ {n}
 +
\int\limits _ { a } ^ { b }  K _ {n} ( x , s ) f ( s) ds .
 +
$$
  
 
The last series is also called a Neumann series. In
 
The last series is also called a Neumann series. In
Line 53: Line 135:
 
the series (3) is considered in the case of an equation (1) to which the Dirichlet problem in potential theory reduces.
 
the series (3) is considered in the case of an equation (1) to which the Dirichlet problem in potential theory reduces.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066440/n06644032.png" /> be a bounded linear operator mapping a Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066440/n06644033.png" /> into itself, with norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066440/n06644034.png" />. Then the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066440/n06644035.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066440/n06644036.png" /> is the identity operator, has a unique bounded inverse <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066440/n06644037.png" />, which admits the expansion
+
Let $  A $
 +
be a bounded linear operator mapping a Banach space $  X $
 +
into itself, with norm $  \| A \| < 1 $.  
 +
Then the operator $  I - A $,  
 +
where $  I $
 +
is the identity operator, has a unique bounded inverse $  ( I - A )  ^ {-} 1 $,  
 +
which admits the expansion
  
<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/n/n066/n066440/n06644038.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
+
$$ \tag{4 }
 +
( I - A )  ^ {-} 1  = \
 +
\sum _ { n= } 0 ^  \infty  A  ^ {n} .
 +
$$
  
 
In the theory of linear operators this series is called a Neumann series. The series
 
In the theory of linear operators this series is called a Neumann series. The series
Line 63: Line 154:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C.G. Neumann,  "Theorie der Besselschen Funktionen" , Teubner  (1867)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  C.G. Neumann,  "Untersuchungen über das logarithmische und Newtonsche Potential" , Teubner  (1877)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G.N. Watson,  "A treatise on the theory of Bessel functions" , '''1''' , Cambridge Univ. Press  (1952)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  R.O. Kuz'min,  "Bessel functions" , Moscow-Leningrad  (1035)  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  K. Yosida,  "Functional analysis" , Springer  (1965)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  F.G. Tricomi,  "Integral equations" , Interscience  (1957)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C.G. Neumann,  "Theorie der Besselschen Funktionen" , Teubner  (1867)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  C.G. Neumann,  "Untersuchungen über das logarithmische und Newtonsche Potential" , Teubner  (1877)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G.N. Watson,  "A treatise on the theory of Bessel functions" , '''1''' , Cambridge Univ. Press  (1952)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  R.O. Kuz'min,  "Bessel functions" , Moscow-Leningrad  (1035)  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  K. Yosida,  "Functional analysis" , Springer  (1965)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  F.G. Tricomi,  "Integral equations" , Interscience  (1957)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
The series (4), applied to a specific vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066440/n06644039.png" />, i.e.
+
The series (4), applied to a specific vector $  f $,  
 +
i.e.
  
<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/n/n066/n066440/n06644040.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
+
$$ \tag{a1 }
 +
\sum _ { n= } 0 ^  \infty  A  ^ {n} f
 +
$$
  
may converge also if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066440/n06644041.png" />. For necessary and sufficient conditions for convergence see [[#References|[a2]]] (or [[#References|[a3]]]).
+
may converge also if $  \| A \| \geq  1 $.  
 +
For necessary and sufficient conditions for convergence see [[#References|[a2]]] (or [[#References|[a3]]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  F. Smithies,  "Integral equations" , Cambridge Univ. Press  (1970)  pp. Chapt. II</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  N. Suzuki,  "On the convergence of Neumann series in Banach space"  ''Math. Ann.'' , '''220'''  (1976)  pp. 143–146</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H.W. Engl,  "A successive-approximation method for solving equations of the second kind with arbitrary spectral radius"  ''J. Integral Eq.'' , '''8'''  (1985)  pp. 239–247</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  I.C. Gohberg,  S. Goldberg,  "Basic operator theory" , Birkhäuser  (1981)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  A.E. Taylor,  D.C. Lay,  "Introduction to functional analysis" , Wiley  (1980)  pp. Chapt. 5</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  P.P. Zabreiko (ed.)  A.I. Koshelev (ed.)  M.A. Krasnoselskii (ed.)  S.G. Mikhlin (ed.)  L.S. Rakovshchik (ed.)  V.Ya. Stet'senko (ed.)  T.O. Shaposhnikova (ed.)  R.S. Anderssen (ed.) , ''Integral equations - a reference text'' , Noordhoff  (1975)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  F. Smithies,  "Integral equations" , Cambridge Univ. Press  (1970)  pp. Chapt. II</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  N. Suzuki,  "On the convergence of Neumann series in Banach space"  ''Math. Ann.'' , '''220'''  (1976)  pp. 143–146</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H.W. Engl,  "A successive-approximation method for solving equations of the second kind with arbitrary spectral radius"  ''J. Integral Eq.'' , '''8'''  (1985)  pp. 239–247</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  I.C. Gohberg,  S. Goldberg,  "Basic operator theory" , Birkhäuser  (1981)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  A.E. Taylor,  D.C. Lay,  "Introduction to functional analysis" , Wiley  (1980)  pp. Chapt. 5</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  P.P. Zabreiko (ed.)  A.I. Koshelev (ed.)  M.A. Krasnoselskii (ed.)  S.G. Mikhlin (ed.)  L.S. Rakovshchik (ed.)  V.Ya. Stet'senko (ed.)  T.O. Shaposhnikova (ed.)  R.S. Anderssen (ed.) , ''Integral equations - a reference text'' , Noordhoff  (1975)  (Translated from Russian)</TD></TR></table>

Revision as of 08:02, 6 June 2020


A series of the form

$$ \sum _ { n= } 0 ^ \infty a _ {n} J _ {\nu + n } ( z) , $$

where $ J _ {\nu + n } $ is the Bessel function (cylinder function of the first kind, cf. Bessel functions) and $ \nu $ is a (real or complex) number. C.G. Neumann

considered the special case when $ \nu $ is an integer. He showed that if $ f ( z) $ is an analytic function in a closed disc with centre at the coordinate origin, $ z $ is an interior point and $ C $ denotes the boundary of the disc, then

$$ f ( z) = \ \sum _ { n= } 0 ^ \infty a _ {n} J _ {n} ( z) , $$

where

$$ a _ {0} = f ( 0) ,\ \ a _ {n} = \frac{1}{\pi i } \int\limits _ { C } O _ {n} ( t) f ( t) dt $$

and $ O _ {n} $ is a polynomial of degree $ n+ 1 $ in $ 1 / t $:

$$ O _ {0} ( t) = \frac{1}{t} , $$

$$ O _ {n} ( t) = \frac{1}{2 t ^ {n+} 1 } \times $$

$$ \times \int\limits _ { 0 } ^ \infty e ^ {-} x [ ( x + \sqrt {x ^ {2} + t ^ {2} } ) ^ {n} + ( x - \sqrt {x ^ {2} + t ^ {2} } ) ^ {n} ] dx ,\ n \geq 1 ; $$

it is usually called the Neumann polynomial of order $ n $. (Neumann himself called it a Bessel function of second order. Nowadays this term is used to denote one of the solutions of the Bessel equation.) Examples of the representation of functions by means of a Neumann series:

$$ \cos ( z \sin \phi ) = \ J _ {0} ( z) + 2 \sum _ { n= } 1 ^ \infty J _ {2n} ( z) \cos 2 n \phi , $$

$$ \sin ( z \sin \phi ) = 2 \sum _ { n= } 1 ^ \infty J _ {2n-} 1 ( z) \sin ( 2n - 1 ) \phi , $$

$$ \left ( { \frac{z}{2} } \right ) ^ \mu = \sum _ { n= } 0 ^ \infty \frac{( \mu + 2n ) \Gamma ( \mu + n ) }{n ! } J _ {\mu + 2n } ( z) , $$

where $ \mu $ is an arbitrary number not equal to a non-negative integer and $ \Gamma $ is the gamma-function.

In the theory of Fredholm integral equations (cf. Fredholm equation)

$$ \tag{1 } \phi ( x) - \lambda \int\limits _ { a } ^ { b } K ( x , s ) \phi ( s) ds = \ f ( x) ,\ \ x \in [ a , b ] , $$

a Neumann series is defined as the expansion of the resolvent $ R ( x , s ; \lambda ) $ of the kernel $ K $:

$$ \tag{2 } R ( x , s ; \lambda ) = \ \sum _ { n= } 1 ^ \infty \lambda ^ {n} K _ {n} ( x , s ) , $$

where the $ K _ {n} $ are the iterated kernels (of $ K $), which are defined by the recurrence formulas

$$ K _ {1} ( x , s ) = K ( x , s ) , $$

$$ K _ {n} ( x , s ) = \int\limits _ { a } ^ { b } K _ {n-} 1 ( x , t ) K ( t , s ) dt ,\ n \geq 2 . $$

By means of (2) the solution of (1) for small $ \lambda $ can be represented by

$$ \tag{3 } \phi ( x) = f ( x) + \sum _ { k= } 1 ^ \infty \lambda ^ {n} \int\limits _ { a } ^ { b } K _ {n} ( x , s ) f ( s) ds . $$

The last series is also called a Neumann series. In

the series (3) is considered in the case of an equation (1) to which the Dirichlet problem in potential theory reduces.

Let $ A $ be a bounded linear operator mapping a Banach space $ X $ into itself, with norm $ \| A \| < 1 $. Then the operator $ I - A $, where $ I $ is the identity operator, has a unique bounded inverse $ ( I - A ) ^ {-} 1 $, which admits the expansion

$$ \tag{4 } ( I - A ) ^ {-} 1 = \ \sum _ { n= } 0 ^ \infty A ^ {n} . $$

In the theory of linear operators this series is called a Neumann series. The series

can be regarded as a special case of (4).

References

[1] C.G. Neumann, "Theorie der Besselschen Funktionen" , Teubner (1867)
[2] C.G. Neumann, "Untersuchungen über das logarithmische und Newtonsche Potential" , Teubner (1877)
[3] G.N. Watson, "A treatise on the theory of Bessel functions" , 1 , Cambridge Univ. Press (1952)
[4] R.O. Kuz'min, "Bessel functions" , Moscow-Leningrad (1035) (In Russian)
[5] K. Yosida, "Functional analysis" , Springer (1965)
[6] F.G. Tricomi, "Integral equations" , Interscience (1957)

Comments

The series (4), applied to a specific vector $ f $, i.e.

$$ \tag{a1 } \sum _ { n= } 0 ^ \infty A ^ {n} f $$

may converge also if $ \| A \| \geq 1 $. For necessary and sufficient conditions for convergence see [a2] (or [a3]).

References

[a1] F. Smithies, "Integral equations" , Cambridge Univ. Press (1970) pp. Chapt. II
[a2] N. Suzuki, "On the convergence of Neumann series in Banach space" Math. Ann. , 220 (1976) pp. 143–146
[a3] H.W. Engl, "A successive-approximation method for solving equations of the second kind with arbitrary spectral radius" J. Integral Eq. , 8 (1985) pp. 239–247
[a4] I.C. Gohberg, S. Goldberg, "Basic operator theory" , Birkhäuser (1981)
[a5] A.E. Taylor, D.C. Lay, "Introduction to functional analysis" , Wiley (1980) pp. Chapt. 5
[a6] P.P. Zabreiko (ed.) A.I. Koshelev (ed.) M.A. Krasnoselskii (ed.) S.G. Mikhlin (ed.) L.S. Rakovshchik (ed.) V.Ya. Stet'senko (ed.) T.O. Shaposhnikova (ed.) R.S. Anderssen (ed.) , Integral equations - a reference text , Noordhoff (1975) (Translated from Russian)
How to Cite This Entry:
Neumann series. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Neumann_series&oldid=47960
This article was adapted from an original article by B.V. Khvedelidze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article