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A mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i0515601.png" /> in which the law of the correspondence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i0515602.png" /> is given by an integral. An integral operator is sometimes called an integral transformation. Thus, for Urysohn's integral operator (cf. [[Urysohn equation|Urysohn equation]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i0515603.png" />, the law of the correspondence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i0515604.png" /> is given by the integral (or the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i0515605.png" /> is generated by the integral)
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<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/i/i051/i051560/i0515606.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
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 +
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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i0515607.png" /> is a given measurable set of finite Lebesgue measure in a finite-dimensional space and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i0515608.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i0515609.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156010.png" />, is a given measurable function. It is assumed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156012.png" /> are functions satisfying conditions that ensure the existence of the integral in (1) in the sense of Lebesgue. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156013.png" /> is a non-linear function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156014.png" />, then (1) is an example of a non-linear integral operator. If, on the other hand, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156015.png" />, then (1) takes the form
+
A mapping  $  x \mapsto A x $
 +
in which the law of the correspondence  $  A $
 +
is given by an integral. An integral operator is sometimes called an integral transformation. Thus, for Urysohn's integral operator (cf. [[Urysohn equation|Urysohn equation]])  $  \phi \mapsto A \phi $,  
 +
the law of the correspondence  $  A $
 +
is given by the integral (or the operator  $  \phi \mapsto A \phi $
 +
is generated by the integral)
  
<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/i/i051/i051560/i05156016.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{1 }
 +
A \phi ( t)  = \int\limits _ { D } P ( t , \tau , \phi ( \tau ) ) d \tau ,\ \
 +
t \in D ,
 +
$$
  
The operator generated by the integral in (2), or simply the operator (2), is called a linear integral operator, and the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156017.png" /> is called its kernel (cf. also [[Kernel of an integral operator|Kernel of an integral operator]]).
+
where  $  D $
 +
is a given measurable set of finite Lebesgue measure in a finite-dimensional space and  $  P ( t , \tau , u ) $,
 +
$  t , \tau \in D $,
 +
$  - \infty < u < \infty $,
 +
is a given measurable function. It is assumed that  $  P $
 +
and  $  \phi $
 +
are functions satisfying conditions that ensure the existence of the integral in (1) in the sense of Lebesgue. If  $  P ( t , \tau , u ) $
 +
is a non-linear function in  $  u $,  
 +
then (1) is an example of a non-linear integral operator. If, on the other hand,  $  P ( t , \tau , u ) = K ( t , \tau ) u $,
 +
then (1) takes the form
  
The kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156018.png" /> is called a [[Fredholm kernel|Fredholm kernel]] if the operator (2) corresponding to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156019.png" /> is completely continuous (compact) from a given function space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156020.png" /> into another function space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156021.png" />. In this case, the operator (2) is called a Fredholm integral operator from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156022.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156023.png" />.
+
$$ \tag{2 }
 +
A \phi ( t) = \int\limits _ { D }
 +
K ( t , \tau ) \phi ( \tau ) d \tau ,\  t \in D .
 +
$$
  
Linear integral operators are often considered in the following function spaces: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156024.png" />, the space of continuous functions on a bounded closed set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156025.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156026.png" />, the space of functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156027.png" /> whose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156028.png" />-th powers are integrable. In the first case (2) is called a [[Fredholm-operator(2)|Fredholm operator]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156029.png" /> (i.e. from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156030.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156031.png" />) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156032.png" /> is continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156033.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156034.png" /> is then called a continuous kernel). The operator (2) is a Fredholm operator in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156035.png" /> (from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156036.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156037.png" />) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156038.png" /> is measurable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156039.png" /> and if
+
The operator generated by the integral in (2), or simply the operator (2), is called a linear integral operator, and the function  $  K $
 +
is called its kernel (cf. also [[Kernel of an integral operator|Kernel of an integral operator]]).
  
<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/i/i051/i051560/i05156040.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
The kernel  $  K $
 +
is called a [[Fredholm kernel|Fredholm kernel]] if the operator (2) corresponding to  $  K $
 +
is completely continuous (compact) from a given function space  $  E $
 +
into another function space  $  E _ {1} $.  
 +
In this case, the operator (2) is called a Fredholm integral operator from  $  E $
 +
into  $  E _ {1} $.
  
Such a kernel is called an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156042.png" />-kernel.
+
Linear integral operators are often considered in the following function spaces:  $  C ( D) $,
 +
the space of continuous functions on a bounded closed set  $  D $,
 +
and  $  L _ {p} ( D) $,
 +
the space of functions on  $  D $
 +
whose  $  p $-
 +
th powers are integrable. In the first case (2) is called a [[Fredholm-operator(2)|Fredholm operator]] in  $  C ( D) $(
 +
i.e. from  $  C( D) $
 +
into  $  C( D) $)
 +
if  $  K $
 +
is continuous on  $  D \times D $(
 +
$  K $
 +
is then called a continuous kernel). The operator (2) is a Fredholm operator in  $  L _ {2} ( D) $(
 +
from  $  L _ {2} ( D) $
 +
into  $  L _ {2} ( D) $)
 +
if  $  K $
 +
is measurable on  $  D \times D $
 +
and if
  
The adjoint of (2), in the complex function space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156043.png" />, with kernel satisfying (3), is the integral operator
+
$$ \tag{3 }
 +
\int\limits _ { D } \int\limits _ { D }
 +
| K ( t , \tau ) |  ^ {2}  d t  d \tau  < \infty .
 +
$$
  
<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/i/i051/i051560/i05156044.png" /></td> <td valign="top" style="width:5%;text-align:right;">(4)</td></tr></table>
+
Such a kernel is called an  $  L _ {2} $-
 +
kernel.
  
where the bar means transition to the complex conjugate. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156045.png" /> is a Hermitian (symmetric) kernel (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156046.png" />), then the corresponding Fredholm operator (2) coincides with its adjoint (4). Operators with this property are called self-adjoint (cf. [[Self-adjoint operator|Self-adjoint operator]]). A Fredholm operator with a symmetric kernel is called a [[Hilbert–Schmidt integral operator|Hilbert–Schmidt integral operator]].
+
The adjoint of (2), in the complex function space  $  L _ {2} ( D) $,
 +
with kernel satisfying (3), is the integral operator
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156047.png" /> denotes the distance between two points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156048.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156049.png" /> in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156050.png" />-dimensional Euclidean space and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156051.png" /> is a bounded measurable function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156052.png" />, then a kernel of the form
+
$$ \tag{4 }
 +
A  ^ {*} \phi ( t)  = \
 +
\int\limits _ { D }
 +
\overline{ {K ( \tau , t ) }}\; \phi ( \tau )  d \tau ,\ \
 +
t \in D ,
 +
$$
  
<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/i/i051/i051560/i05156053.png" /></td> <td valign="top" style="width:5%;text-align:right;">(5)</td></tr></table>
+
where the bar means transition to the complex conjugate. If  $  K $
 +
is a Hermitian (symmetric) kernel (i.e.  $  \overline{ {K ( \tau , t ) }}\; = K ( t , \tau ) $),
 +
then the corresponding Fredholm operator (2) coincides with its adjoint (4). Operators with this property are called self-adjoint (cf. [[Self-adjoint operator|Self-adjoint operator]]). A Fredholm operator with a symmetric kernel is called a [[Hilbert–Schmidt integral operator|Hilbert–Schmidt integral operator]].
 +
 
 +
If  $  | t - \tau | $
 +
denotes the distance between two points  $  t $
 +
and  $  \tau $
 +
in the  $  n $-
 +
dimensional Euclidean space and if  $  B ( t , \tau ) $
 +
is a bounded measurable function on  $  D \times D $,
 +
then a kernel of the form
 +
 
 +
$$ \tag{5 }
 +
K ( t , \tau )  = \
 +
 
 +
\frac{B ( t , \tau ) }{| t - \tau |  ^ {m} }
 +
,\ \
 +
0 < m < n ,
 +
$$
  
 
is called a kernel of potential type, and an operator (2) with such a kernel is called an integral operator of potential type. The kernel (5) is also called a polar kernel, or a kernel with weak singularity, while the corresponding operator (2) is called an integral operator with weak singularity.
 
is called a kernel of potential type, and an operator (2) with such a kernel is called an integral operator of potential type. The kernel (5) is also called a polar kernel, or a kernel with weak singularity, while the corresponding operator (2) is called an integral operator with weak singularity.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156054.png" /> is a continuous function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156055.png" />, the corresponding integral operator with weak singularity is completely continuous in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156056.png" />, while if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156057.png" /> is a bounded measurable function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156058.png" />, the corresponding operator is completely continuous in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156059.png" />.
+
If $  B ( t , \tau ) $
 +
is a continuous function on $  D \times D $,  
 +
the corresponding integral operator with weak singularity is completely continuous in $  C ( D) $,  
 +
while if $  B ( t , \tau ) $
 +
is a bounded measurable function on $  D \times D $,  
 +
the corresponding operator is completely continuous in $  L _ {2} ( D) $.
  
If the kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156060.png" /> and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156061.png" />-dimensional set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156062.png" /> are such that the integral (2) does not exist in the sense of Lebesgue, but does exist in the sense of the Cauchy principal value, then the integral (2) is called an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156064.png" />-dimensional singular integral (cf. also [[Singular integral|Singular integral]]). The operator generated by it is called an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156066.png" />-dimensional singular integral operator, or a one-dimensional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156067.png" /> or multi-dimensional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156068.png" /> singular integral operator.
+
If the kernel $  K $
 +
and the $  m $-
 +
dimensional set $  D $
 +
are such that the integral (2) does not exist in the sense of Lebesgue, but does exist in the sense of the Cauchy principal value, then the integral (2) is called an $  m $-
 +
dimensional singular integral (cf. also [[Singular integral|Singular integral]]). The operator generated by it is called an $  m $-
 +
dimensional singular integral operator, or a one-dimensional $  ( m = 1 ) $
 +
or multi-dimensional $  ( m > 1 ) $
 +
singular integral operator.
  
If a curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156069.png" /> lies in the complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156070.png" />-plane, then
+
If a curve $  D $
 +
lies in the complex $  t $-
 +
plane, 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/i/i051/i051560/i05156071.png" /></td> <td valign="top" style="width:5%;text-align:right;">(6)</td></tr></table>
+
$$ \tag{6 }
 +
A \phi ( t) = \
 +
\int\limits _ { D }
  
where the integral is understood in the sense of the Cauchy principal value, generates a continuous integral operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156072.png" /> in the space of functions satisfying a Hölder condition (if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156073.png" /> is a simple closed curve) or in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156074.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156075.png" /> (if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156076.png" /> is a Lyapunov curve, cf. also [[Lyapunov surfaces and curves|Lyapunov surfaces and curves]]). The operator (6) is called a singular Cauchy operator.
+
\frac{\phi ( \tau ) }{\tau - t }
 +
\
 +
d \tau ,\  t \in D ,
 +
$$
  
Suppose that two Lebesgue-measurable functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156077.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156078.png" /> are given on the real axis. If for almost-all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156079.png" /> the integral
+
where the integral is understood in the sense of the Cauchy principal value, generates a continuous integral operator  $  \phi \mapsto A \phi $
 +
in the space of functions satisfying a Hölder condition (if  $  D $
 +
is a simple closed curve) or in  $  L _ {p} ( D) $,
 +
$  1 < p < \infty $(
 +
if  $  D $
 +
is a Lyapunov curve, cf. also [[Lyapunov surfaces and curves|Lyapunov surfaces and curves]]). The operator (6) is called a singular Cauchy operator.
  
<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/i/i051/i051560/i05156080.png" /></td> </tr></table>
+
Suppose that two Lebesgue-measurable functions  $  g $
 +
and  $  \phi $
 +
are given on the real axis. If for almost-all  $  t \in ( - \infty , \infty ) $
 +
the integral
 +
 
 +
$$
 +
\int\limits _ {- \infty } ^  \infty 
 +
| \phi ( \tau ) |  | g ( \tau - t ) |  d \tau
 +
$$
  
 
exists, then one can define the function
 
exists, then one can define the function
  
<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/i/i051/i051560/i05156081.png" /></td> <td valign="top" style="width:5%;text-align:right;">(7)</td></tr></table>
+
$$ \tag{7 }
 +
( g \star \phi ) ( t)  = \
 +
\int\limits _ {- \infty } ^  \infty 
 +
g ( \tau - t ) \phi ( \tau ) d \tau ,
 +
$$
  
called the convolution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156082.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156083.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156084.png" /> is fixed, (7) defines an operator
+
called the convolution of $  g $
 +
and $  \phi $.  
 +
If $  g $
 +
is fixed, (7) defines an operator
  
<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/i/i051/i051560/i05156085.png" /></td> <td valign="top" style="width:5%;text-align:right;">(8)</td></tr></table>
+
$$ \tag{8 }
 +
T \phi ( t)  = ( g \star \phi ) ( t) ,
 +
$$
  
which is called the convolution integral operator (or convolution integral transform, cf. also [[Convolution transform|Convolution transform]]) with kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156086.png" />.
+
which is called the convolution integral operator (or convolution integral transform, cf. also [[Convolution transform|Convolution transform]]) with kernel $  g $.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156087.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156088.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156089.png" />, then (8) is a continuous operator from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156090.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156091.png" />. Under corresponding conditions, convolution integral operations are used on a semi-axis as well as on a finite interval.
+
If $  g \in L _ {r} ( - \infty , \infty ) $,
 +
$  1 \leq  p , q , r \leq  \infty $,  
 +
and $  ( 1 / p ) - ( 1 / q ) = ( 1 / r ) - 1 $,  
 +
then (8) is a continuous operator from $  L _ {q} ( - \infty , \infty ) $
 +
into $  L _ {p} ( - \infty , \infty ) $.  
 +
Under corresponding conditions, convolution integral operations are used on a semi-axis as well as on a finite interval.
  
 
Apart from the integral operators above, concrete classes of integral operators have been studied, e.g. the integral transforms of Fourier, Laplace, Bessel, Mellin, Hilbert, etc.
 
Apart from the integral operators above, concrete classes of integral operators have been studied, e.g. the integral transforms of Fourier, Laplace, Bessel, Mellin, Hilbert, etc.
Line 59: Line 183:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.I. Smirnov,  "A course of higher mathematics" , '''5''' , Addison-Wesley  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</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><TR><TD valign="top">[3]</TD> <TD valign="top">  M.A. Krasnosel'skii,  et al.,  "Integral operators in spaces of summable functions" , Noordhoff  (1976)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.A. Ditkin,  A.P. Prudnikov,  "Integral transforms and operational calculus" , Pergamon  (1965)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  N.I. Muskhelishvili,  "Singular integral equations" , Wolters-Noordhoff  (1953)  (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  S.G. Mikhlin,  "Multi-dimensional singular integrals and integral equations" , Pergamon  (1965)  (Translated from Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  E.C. Titchmarsh,  "Introduction to the theory of Fourier integrals" , Oxford Univ. Press  (1948)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  R.E. Edwards,  "Functional analysis: theory and applications" , Holt, Rinehart &amp; Winston  (1965)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.I. Smirnov,  "A course of higher mathematics" , '''5''' , Addison-Wesley  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</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><TR><TD valign="top">[3]</TD> <TD valign="top">  M.A. Krasnosel'skii,  et al.,  "Integral operators in spaces of summable functions" , Noordhoff  (1976)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  V.A. Ditkin,  A.P. Prudnikov,  "Integral transforms and operational calculus" , Pergamon  (1965)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  N.I. Muskhelishvili,  "Singular integral equations" , Wolters-Noordhoff  (1953)  (Translated from Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  S.G. Mikhlin,  "Multi-dimensional singular integrals and integral equations" , Pergamon  (1965)  (Translated from Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  E.C. Titchmarsh,  "Introduction to the theory of Fourier integrals" , Oxford Univ. Press  (1948)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  R.E. Edwards,  "Functional analysis: theory and applications" , Holt, Rinehart &amp; Winston  (1965)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I.C. Gohberg,  S. Goldberg,  "Basic operator theory" , Birkhäuser  (1981)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P.R. Halmos,  V.S. Sunder,  "Bounded integral operators on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156092.png" /> spaces" , Springer  (1978)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  K. Jörgens,  "Lineare Integraloperatoren" , Teubner  (1970)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A.C. Zaanen,  "Linear analysis" , North-Holland  (1956)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I.C. Gohberg,  S. Goldberg,  "Basic operator theory" , Birkhäuser  (1981)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P.R. Halmos,  V.S. Sunder,  "Bounded integral operators on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051560/i05156092.png" /> spaces" , Springer  (1978)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  K. Jörgens,  "Lineare Integraloperatoren" , Teubner  (1970)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A.C. Zaanen,  "Linear analysis" , North-Holland  (1956)</TD></TR></table>

Revision as of 22:12, 5 June 2020


A mapping $ x \mapsto A x $ in which the law of the correspondence $ A $ is given by an integral. An integral operator is sometimes called an integral transformation. Thus, for Urysohn's integral operator (cf. Urysohn equation) $ \phi \mapsto A \phi $, the law of the correspondence $ A $ is given by the integral (or the operator $ \phi \mapsto A \phi $ is generated by the integral)

$$ \tag{1 } A \phi ( t) = \int\limits _ { D } P ( t , \tau , \phi ( \tau ) ) d \tau ,\ \ t \in D , $$

where $ D $ is a given measurable set of finite Lebesgue measure in a finite-dimensional space and $ P ( t , \tau , u ) $, $ t , \tau \in D $, $ - \infty < u < \infty $, is a given measurable function. It is assumed that $ P $ and $ \phi $ are functions satisfying conditions that ensure the existence of the integral in (1) in the sense of Lebesgue. If $ P ( t , \tau , u ) $ is a non-linear function in $ u $, then (1) is an example of a non-linear integral operator. If, on the other hand, $ P ( t , \tau , u ) = K ( t , \tau ) u $, then (1) takes the form

$$ \tag{2 } A \phi ( t) = \int\limits _ { D } K ( t , \tau ) \phi ( \tau ) d \tau ,\ t \in D . $$

The operator generated by the integral in (2), or simply the operator (2), is called a linear integral operator, and the function $ K $ is called its kernel (cf. also Kernel of an integral operator).

The kernel $ K $ is called a Fredholm kernel if the operator (2) corresponding to $ K $ is completely continuous (compact) from a given function space $ E $ into another function space $ E _ {1} $. In this case, the operator (2) is called a Fredholm integral operator from $ E $ into $ E _ {1} $.

Linear integral operators are often considered in the following function spaces: $ C ( D) $, the space of continuous functions on a bounded closed set $ D $, and $ L _ {p} ( D) $, the space of functions on $ D $ whose $ p $- th powers are integrable. In the first case (2) is called a Fredholm operator in $ C ( D) $( i.e. from $ C( D) $ into $ C( D) $) if $ K $ is continuous on $ D \times D $( $ K $ is then called a continuous kernel). The operator (2) is a Fredholm operator in $ L _ {2} ( D) $( from $ L _ {2} ( D) $ into $ L _ {2} ( D) $) if $ K $ is measurable on $ D \times D $ and if

$$ \tag{3 } \int\limits _ { D } \int\limits _ { D } | K ( t , \tau ) | ^ {2} d t d \tau < \infty . $$

Such a kernel is called an $ L _ {2} $- kernel.

The adjoint of (2), in the complex function space $ L _ {2} ( D) $, with kernel satisfying (3), is the integral operator

$$ \tag{4 } A ^ {*} \phi ( t) = \ \int\limits _ { D } \overline{ {K ( \tau , t ) }}\; \phi ( \tau ) d \tau ,\ \ t \in D , $$

where the bar means transition to the complex conjugate. If $ K $ is a Hermitian (symmetric) kernel (i.e. $ \overline{ {K ( \tau , t ) }}\; = K ( t , \tau ) $), then the corresponding Fredholm operator (2) coincides with its adjoint (4). Operators with this property are called self-adjoint (cf. Self-adjoint operator). A Fredholm operator with a symmetric kernel is called a Hilbert–Schmidt integral operator.

If $ | t - \tau | $ denotes the distance between two points $ t $ and $ \tau $ in the $ n $- dimensional Euclidean space and if $ B ( t , \tau ) $ is a bounded measurable function on $ D \times D $, then a kernel of the form

$$ \tag{5 } K ( t , \tau ) = \ \frac{B ( t , \tau ) }{| t - \tau | ^ {m} } ,\ \ 0 < m < n , $$

is called a kernel of potential type, and an operator (2) with such a kernel is called an integral operator of potential type. The kernel (5) is also called a polar kernel, or a kernel with weak singularity, while the corresponding operator (2) is called an integral operator with weak singularity.

If $ B ( t , \tau ) $ is a continuous function on $ D \times D $, the corresponding integral operator with weak singularity is completely continuous in $ C ( D) $, while if $ B ( t , \tau ) $ is a bounded measurable function on $ D \times D $, the corresponding operator is completely continuous in $ L _ {2} ( D) $.

If the kernel $ K $ and the $ m $- dimensional set $ D $ are such that the integral (2) does not exist in the sense of Lebesgue, but does exist in the sense of the Cauchy principal value, then the integral (2) is called an $ m $- dimensional singular integral (cf. also Singular integral). The operator generated by it is called an $ m $- dimensional singular integral operator, or a one-dimensional $ ( m = 1 ) $ or multi-dimensional $ ( m > 1 ) $ singular integral operator.

If a curve $ D $ lies in the complex $ t $- plane, then

$$ \tag{6 } A \phi ( t) = \ \int\limits _ { D } \frac{\phi ( \tau ) }{\tau - t } \ d \tau ,\ t \in D , $$

where the integral is understood in the sense of the Cauchy principal value, generates a continuous integral operator $ \phi \mapsto A \phi $ in the space of functions satisfying a Hölder condition (if $ D $ is a simple closed curve) or in $ L _ {p} ( D) $, $ 1 < p < \infty $( if $ D $ is a Lyapunov curve, cf. also Lyapunov surfaces and curves). The operator (6) is called a singular Cauchy operator.

Suppose that two Lebesgue-measurable functions $ g $ and $ \phi $ are given on the real axis. If for almost-all $ t \in ( - \infty , \infty ) $ the integral

$$ \int\limits _ {- \infty } ^ \infty | \phi ( \tau ) | | g ( \tau - t ) | d \tau $$

exists, then one can define the function

$$ \tag{7 } ( g \star \phi ) ( t) = \ \int\limits _ {- \infty } ^ \infty g ( \tau - t ) \phi ( \tau ) d \tau , $$

called the convolution of $ g $ and $ \phi $. If $ g $ is fixed, (7) defines an operator

$$ \tag{8 } T \phi ( t) = ( g \star \phi ) ( t) , $$

which is called the convolution integral operator (or convolution integral transform, cf. also Convolution transform) with kernel $ g $.

If $ g \in L _ {r} ( - \infty , \infty ) $, $ 1 \leq p , q , r \leq \infty $, and $ ( 1 / p ) - ( 1 / q ) = ( 1 / r ) - 1 $, then (8) is a continuous operator from $ L _ {q} ( - \infty , \infty ) $ into $ L _ {p} ( - \infty , \infty ) $. Under corresponding conditions, convolution integral operations are used on a semi-axis as well as on a finite interval.

Apart from the integral operators above, concrete classes of integral operators have been studied, e.g. the integral transforms of Fourier, Laplace, Bessel, Mellin, Hilbert, etc.

References

[1] V.I. Smirnov, "A course of higher mathematics" , 5 , Addison-Wesley (1964) (Translated from Russian)
[2] 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)
[3] M.A. Krasnosel'skii, et al., "Integral operators in spaces of summable functions" , Noordhoff (1976) (Translated from Russian)
[4] V.A. Ditkin, A.P. Prudnikov, "Integral transforms and operational calculus" , Pergamon (1965) (Translated from Russian)
[5] N.I. Muskhelishvili, "Singular integral equations" , Wolters-Noordhoff (1953) (Translated from Russian)
[6] S.G. Mikhlin, "Multi-dimensional singular integrals and integral equations" , Pergamon (1965) (Translated from Russian)
[7] E.C. Titchmarsh, "Introduction to the theory of Fourier integrals" , Oxford Univ. Press (1948)
[8] R.E. Edwards, "Functional analysis: theory and applications" , Holt, Rinehart & Winston (1965)

Comments

References

[a1] I.C. Gohberg, S. Goldberg, "Basic operator theory" , Birkhäuser (1981)
[a2] P.R. Halmos, V.S. Sunder, "Bounded integral operators on spaces" , Springer (1978)
[a3] K. Jörgens, "Lineare Integraloperatoren" , Teubner (1970)
[a4] A.C. Zaanen, "Linear analysis" , North-Holland (1956)
How to Cite This Entry:
Integral operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Integral_operator&oldid=15387
This article was adapted from an original article by B.V. Khvedelidze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article